Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/courseinmathemat01gouruoft A COURSE IN MATHEMATICAL ANALYSIS BY 6D0UARD GOURSAT "PBOwnaaoH op Mathematics in the Univeiwity ok Paris TRANSLATED BY EARLE RAYMOND HEDRICK Pbofessob op Mathematics in the University ok Missouri Vol. I DERIVATIVES AND DIFFERENTIALS DEFINITE INTEGRALS EXPANSION IN SERIES APPLICATIONS TO GEOMETRY GINN AND COMPANY BOSTON • NKW YORK • CHICAOO • LONDON ATI AM A DALLAS • COLl'MBUS • SAN FHANCISCO 3)o3 V.J ENTEBBD AT STATIONERS* HALL COPTBIGHT, 1904, BY EARLE RAYMOND HEDRICK ALL RIGHTS RESERVED PBIKTED IN THE UNITED STATES OF AMERICA 436.4 CINN ANU COMPANY • PRO- PBIBTOM . MtTON • MAJk. AUTHOR'S PREFACE This book contains, with slight variations, the material given in my course at the University of Paris. I have modified somewhat the order fullowed in the lectures for the sake of uniting in a single volume all that has to do with functions of real variables, except the theory of differential equations. The differential notation not being treated in the ** Classe de Math^matiques sp^iales," ♦ I have treated this notation from the beginning, and have presupposed only a knowledge of the formal rules for calculating derivatives. Since mathematical analysis is essentially the science of the con- tinuum, it would seem that every course in analysis should begin, logically, with the study of irrational numbers. I have supposed, however, that the student is already familiar with that subject. The theory of incoiiimensurable numbers is treated in so many excellent well-known works f that I have thought it useless to enter upon such a discussion. As for the other fundamental notions which lie at the basis of analysis, — such as the upper limit, the definite integral, the double integral, etc., — I have endeavored to treat them with all desirable rigor, seeking to retain the elementary character of the work, and to avoid generalizations which would be superfluous in a book intended for purposes of instruction. Certain paragraphs which are printed in smaller type than the body of the book contain either problems solved in detail or else • An iDteresting account of French methods of instruction in mathematics will be found in an article by Pierpont, BuUetin Amer. Math. Society, Vol. VI, 2d series (1900), p. 225.— Trans. t Such books are not common in En^jlish. The reader is referred to Pierpont, Theory of Functions of Real I'ariahlefi, Ginn & Comimny, Boston. 1905; Tanner>*, Lerofus d'arithmHique, 1900, and other foreign works on arithmetic and on real functions. ill iT AUTHOR'S PREFACE supplementary matter which the reader may omit at the first read- ing without inconvenience. Each chapter is followed by a list of exjunplei which are directly illustrative of the methods treated in the chapter. Most of these examples have been set in examina- tions. Certain others, which are designated by an asterisk, are •omewhat more difficult The latter are taken, for the most part, from original memoirs to which references are made. Two of my old students at the ^^cole Normale, M. ^^mile Cotton and }L Jean Clairin, have kindly assisted in the correction of proofs ; I take this occasion to tender them my hearty thanks. J«o«,87.1»(« B.GOUBSAT TRANSLATOR'S PREFACE The translation of this Course was undertaken at the suggestion of Professor W. F. Osgood, whose review of the original appeared in the July number of the Bulletin of the American Mathematical Society in 1903. The lack of standard texts on mathematical sub- jects in the English language is too well known to require insistence. I earnestly hope that this book will help to fill the need so generally felt throughout the American mathematical world. It may be used conveniently in our system of instruction as a text for a second course in calculus, and as a book of reference it will be found valuable to an American student throughout his work. Few alterations have been made from the French text. Slight changes of notation have been introduced occasionally for conven- ience, and several changes and additions have been made at the sug- gestion of Professor Goursat, who has very kindly interested himself in the work of translation. To him is due all the additional matter not to be found in the French text, except the footnotes which are signed, and even these, though not of his initiative, were always edited by him. I take this opportunity to express my gratitude to the author for the permission to translate the work and for the sympathetic attitude which he has consistently assumed. I am also indebted to Professor Osgood for counsel as the work progressed and for aid in doubtful matters pertaining to the translation. The publishers, Messrs. Ginn «& Company, have spared no pains to make the typography excellent. Their spirit has been far from com- mercial in the whole enterprise, and it is their hope, as it is mine, that the publication of this book will contribute to the advance of mathematics in America. ^ ^ HEDRICK August, 1004 CONTENTS Chaitbr Paob I. Derivatives and Differentials 1 I. Functions of a Single Variable 1 II. Functions of Several Variables . . .11 III. The Differential Notation 10 II. Implicit Functions. Functional Determinants. Change OF Variable 35 I. Implicit Functions 35 II. Functional Determinants 52 III. Transformations ... .... 61 III. Taylor's Series. Elementary Applications. Maxima and Minima I. Taylor's Series with a Remainder. Taylor's Series II. Singular Points. Maxima and Minima . IV. Definite Integrals I. Special Methods of Quadrature .... II. Definite Integrals. Allied Geometrical Concepts . III. Change of Variable. Integration by Parts IV. Generalizations of the Idea of an Integral. Improper Integrals. Line Integrals V. Functions defined by Definite Integrals . VI. Approximate Evaluation of Definite Integrals V. Indefinite Integrals I. Integration of Rational Functions .... II. Elliptic and Ilyperelliptic Integrals III. Integration of Transcendental Functions VL Double Integrals I. Double Integrals. Methinls of Evaluation. (Jr»'»'U*s Theorem . . n. Change of Variable.s. An-a oi a Miria««' III. Generalization.s of Double Integrals. Improper Int<»grals Surface Integrals ...... IV. Analytical and Geometrical Applications vil 80 110 134 134 140 166 175 192 106 208 208 L>>6 236 250 250 264 284 Yiii CONTENTS Paox VIL MVLTIPLB IlTTBORALS. IhTEORATION OF TOTAL DlFFER- EKTIALS 296 I. Multiple IntegralB. Change of Variables .296 n. iBtagration of Totel Differentials 813 niL IxrufiTK SsRiES 327 L Series of Real Constant Terms. General Properties. Tests for Convergence . . . 327 n. Series of Complex Terms. Multiple Series 350 HL Series of Variable Terms. Uniform Convergence . . 860 IX. POWRR SkRIES. TRIOOirOMETRFC SERIES .... 875 I. Power Series of a Single Variable 875 n. Power Series in Several Variables 894 in. Implicit Functions. Analytic Curves and Surfaces . 399 IV. Trigonometric Series. Miscellaneous Series . . .411 X. Plane Curves 426 I. Envelopes 426 n. Curvature 433 ni. Contact of Plane Curves 448 XI. Skew CrRVEs 453 I. Osculating Plane 453 II. Envelopes of Surfaces 459 III. Curvature and Torsion of Skew Curves .... 468 IV. Contact between Skew Curves. Contact between Curves and Surfaces 486 Xn. BuwwAcms 497 I. Curvature of Curves drawn on a Surface . 497 n. Asjrmptotic Lines. Conjugate Lines .... 506 in. Lines of Curvature 514 IV. Families of Straight Lines 526 '>"»»» 541 I A COURSE m MATHEMATICAL ANALYSIS CHAPTER I DERIVATIVES AND DIFFERENTIALS L FUNCTIONS OF A SINGLE VARIABLE 1. Limits. When the successive vahies of a variable x approach nearer and nearer a constant quantity a, in such a way that the absolute value of the difference x — a finally becomes and remains less than any preassigned number, the constant a is called the limit of the variable x. This definition furnishes a criterion for determining whether a is the limit of the variable x. The neces- sary and sufficient condition that it should be, is that, given any positive number «, no matter how small, the absolute value of x — a should remain less than < for all values which the variable x can assume, after a certain instant. Numerous examples of limits are to be found in Geometry and Algebra. For example, the limit of the variable quantity 05 = (a' — m^) / (a — /w), as m approaches a, is 2 a ; for x — 2a will be less than e whenever m — a is taken less than e. Likewise, the variable x = a — 1 /n, where w is a positive integer, approaches the limit a when n increases indefinitely ; for a — a* is less than c when- ever n is greater than l/«. It is apparent from these examples that the successive values of the variable x, as it approaches its limit, may form a continuous or a discontinuous sequence. It is in general very difficult to determine the limit of a variable quantity. The following proposition, which we will assume as self- evident, enables us, in many cases, to establish the existence of a limit Any variable quantity which nether decreases, and which always remains less than a constant quantity L, approaches a limit I, which is less than or at most equal to L. Similarly, any variable quantity which never increases^ and whieh always remains greater than a constant quantity L\ approaches a limit I'f which is greater than or else equal to L\ 1 S DERIVATIVES AND DIFFERENTIALS [I, §2 ''For example, if each of an infinite series of positive terms is leM, wspeotiTely, than the corresponding term of another infinite •eriM of positiYe terms which is known to converge, then the first mrim oooTerges also ; for the sum 2, of the first n terms evidently incieasM with n, and this sum is constantly less than the total sum S of the second series. 2. FuncUont. When two variable quantities are so related that the ralue of one of them depends upon the value of the other, they are said to be functions of each other. If one of them be sup- posed to vary arbitrarily, it is called the independent variable. Let this rariable be denoted by a:, and let us suppose, for example, that it can assume all values between two given numbers a and b (a < b). Let y be another variable, such that to each value of x b e t w e en a and 6, and also for the values a and b themselves, there oorresponds one definitely determined value of y. Then y is called a function of ar, defined in the interval (a, b) ; and this dependence is indicated by writing the equation y —f(x). For instance, it may happen that y is the result of certain arithmetical operations per- formed upon X. Such is the case for the very simplest fimctions studied in elementary mathematics, e.g. polynomials, rational func- tions, radicals, etc. A function may also be defined graphically. Let two coordinate axes OXf Oy be taken in a plane ; and let us join any two points A and B of this plane by a curvilinear arc A CB, of any shape, which is not cut in more than one point by any parallel to the axis Oy. Then the ordinate of a point of this curve will be a function of the a bsc i ssa. The arc A CB may be composed of several distinct por- tions which belong to'&ifferent curves, such as segments of straight lines, arcs of circles, etc. In short, any absolutely arbitrary law may be assumed for finding the ralue of y from that of x. The word function, in its most gen- tial sense, means nothing more nor less than this : to every value of s eonesponds a value of y. 8. Continnity. The definition of functions to which the infini- tesimal oaloulus applies does not admit of such broad generality. Lai y -/(as) be a function d%gned in a certain interval (a, 6), and let «, and «t + A be two values of x in that interval. If the differ- /('• + *)-/(«o) approaches zero as the absolute value of h sro, the function /(x) is said to be continuous for the m^ From the very definition of a limit we may also say that 1,53] FUNCTIONS OF A SINGLE VARIABLE 8 a function f{x) is continuous for x = Xo if, corresponding to every positive number «, no matter how smcUly we can find a positive num^ ber rii such that |/(x. + A)-/(x„)|<. for every value of h less than rj in absolute value.* We shall say that a function f(x) is continuous in an interval (a, b) if it is continuous for every value of x lying in that interval, and if the differences /(a + A)-/(a), fib -h)- fib) each approach zero when A, which is now to be taken only positive, approaches zero. Irt elementary text-books it is usually shown that polynomials, rational functions, the exponential and the logarithmic function, the trigonometric functions, and the inverse trigonometric functions are continuous functions, except for certain particular values of the variable. It follows directly from the definition of continuity that the sum or the product of any number of continuous functions is itself a continuous function ; and this holds for the quotient of two continuous functions also, except for the values of the variable for which the denominator vanishes. It seems superfluous to explain here the reasons which lead us to assume that functions which are defined by physical conditions are, at least in general, continuous. Among the properties of continuous functions we shall now state only the two following, which one might be tempted to think were self-evident, but which really amount to actual theorems, of which rigorous demonstrations will be given later, f I. If the function y =f(x) is continuous in the interval (a, ft), and if N is a number between f (a) a7idfib)y then the equation f(x) = N has at least one root between a and b. II. There exists at least one value of x belonging to the interval (a, b), inclusii^e of its end points^ for which y takes on a value M which is greater than, or at least equal to, the value of the function at any other point in the interval. Likewise, there exists a value of x for which y takes on a value m, than which the function assumes no smaller value in the interval. The numbers M and m are called the maximum and the minimum values of f(x), respectively, in the interval (o, h). It is clear th^ * The notation \ a | denotes the absolute value of a. t See Chapter IV. 4 I>BRIVATrVES AND DIFFERENTIALS [I, §4 the ralue of x tat which /(x) assumes its maximum value AT, or the ralne of x oonesponding to the minimum m, may be at one of the end points, a or 6. It follows at once from the two theorems above, that if iV is a number between M and m, the equation f{x) = N has at least one root which lies between a and h. 4L Examples of discontinuities. The functions which we shall study will be in general continuous, but they may cease to be so for certain exceptional values of the variable. We proceed to give several examples of the kinds of discontinuity which occur most frequently. The function y = l/(x — a) is continuous for every value Xq of X except a. The operation necessary to determine the value of y frt>m that of X ceases to have a meaning when x is assigned the value a ; but we note that when x is very near to a the absolute value of y is very large, and y is positive or negative with x — a. Ab the difference x — a diminishes, the absolute value of y increases indefinitely, so as eventually to become and remain greater than any praasaigned number. This phenomenon is described by saying that jf beeomes infinite when x = a. Discontinuity of this kind is of great importance in Analysis. Let us consider next the function y = sin 1 /x. As x approaches xero, 1 /x increases indefinitely, and y does not approach any limit whatever, although it remains between + 1 and - 1. The equation nnl/x = Ay where|^|<l, has an infinite number of solutions which lie between and c, no matter how small e be taken. What- ever value be assigned to y when a; = 0, the function under con- sideration cannot be made continuous for x = 0. An example of a still different kind of discontinuity is given by the convergent infinite series "(^> = ^"^iT^-^ • + (1 + xy When • approaches zero, S(x) approaches the limit 1, although 5(0) mm 0. For, when a: = 0, every term of the series is zero, and heooo S(0) = 0. But if a; be given a value different from zero, a geometric progression is obtained, of which the ratio is 1/(1 -|- x^ 8(x) f!_ = fl(l±f3-i .,.. I.§a] FLT^CTIONS OF A SINGLE VARIABLE 5 and the limit of S(x) is seen to be 1. Thus, in this example, the function approaches a delinite limit as x approaches zero, but that limit is different from the value of the function for x = 0. 5. Derivatives. Let/(a:) be a continuous function. Then the two terms of the quotient /(x + A)-/(ar) A approach zero simultaneously, as the absolute value of h approaches zero, while x remains fixed. If this quotient approaches a limit, this limit is called the derivative of the function /(a*), and is denoted by I/', or by /' (x), in the notation due to Lagrange. An important geometrical concept is associated with this analytic notion of derivative. Let us consider, in a plane XOY, the curve A MB, which represents the function y =/(x), which we shall assume to be continuous in the interval (a, b). Let M and M' be two points ' on this curve, in the interval (a, b), and let their abscissai be x and X 4- A, respectively. The slope of the straight line MM' is then precisely the quotient above. Now as h approaches zero the point M' approaches the point 3/; and, if the function has a derivative, the slo|)e of the line MM' approaches the limit y'. The straight line MM', therefore, approaches a limiting position, which is called the tangent to the curve. It follows that the equation of the tangent is Y-y = y'{X-x), where X and Y are the running coordinates. To generalize, let us consider any curve in space, and let be the coordinates of a point on the curve, expressed as functions of a variable parameter t. Let M and M' be two points of the curve corresponding to two values, t and t -\- h, of the parameter. The equations of the chord MM' are then X-f(f) ^ Y-<t>(f) ^ Z-^(t) f{t + h) -f(t) <i>{t + h) - <l>{t) ^(t 4- h) - ^(t) If we divide each denominator by h and then let h approach zero, the chord Myf evidently approaches a limiting position, which is given by the equations fit) <t>'(t) ^'(t) ' 6 DERIVATIVES AND DIFFERENTIALS [I, §5 proTided, of course, that each of the three functions /(O, <^(0» ^(0 potMtaes a derivative. The determination of the tangent to a curve thus wducea, analytically, to the calculation of derivatives. Every function which powesses a derivative is necessarily con- tinuous, but the converse is not true. It is easy to give examples of continuous functions which do not possess derivatives for par- ticular values of the variable. The function y = xsinl/a;, for example, is a perfectly continuous function of a, for x = 0,* and y approacherzero as x approaches zero. But the ratio y jx = sinl/o; does not approach any limit whatever, as we have already seen. Let us next consider the function y = xl Here y is continuous for every value of a; ; and y = when a; = 0. But the ratio y/x=^x~^ increases indefinitely as x approaches zero. For abbreviation the derivative is said to be infinite for x = ; the curve which repre- sents the function is tangent to the axis of y at the origin. Finally, the function xe^ y = i is continuous at x = 0,* but the ratio y /x approaches two different limits according as a; is always positive or always negative while it is approaching zero. When x is positive and small, e^/"" is posi- tive and very large, and the ratio y /x approaches 1. But if x is negative and very small in absolute value, e^^^ is very small, and the ratio y /x approaches zero. There exist then two values of the derivative according to the manner in which x approaches zero : the curve which represents this function has a comer at the origin. It is clear from these examples that there exist continuous func- tions which do not possess derivatives for particular values of the variable. But the discoverers of the infinitesimal calculus confi- dently believed that a continuous function had a derivative in gen- eral. Attempts at proof were even made, but these were, of course, fallacious. Finally, Weierstrass succeeded in settling the question conclusively by giving examples of continuous functions which do not possess derivatives for any values of the variable whatever.f But as these functions hseve not as yet been employed in any applications, * AflUr the valne laro hat been auigned to y for x = C — Translator. t Note tmd at the Academy of Science! of Berlin, July 18, 1872. Other examples are to ba foaad la the memoir by Darboux on discoutinuous functions (Annales de finU NcrmaU SupHimtn, Vol. IV, 'id series). One of Weierstrass's examples is flvMi later (Chapter IX). 1,56] FUNCTIONS OF A SINGLE VARIABLE 7 we shall not consider them here. In the future, when we say that a function f{x) has a derivative in the interval (a, 6), we shall mean that it has an unique Jinite derivative for every value of x between '/ jiiid b and also for x = a (A being positive) and for x = b (h being negative), unless an explicit statement is made to the contrary. 6. Successive derivatives. The derivative of a function f(x) is in general another function of x,f'(x). U f'(x) in turn has a deriva- tive, the new function is called the second derivative of /^, and is represented by y" or by f"(x). In the same way the third deriva- tive I/'", or f"'{x), is defined to be the derivative of the second, and so on. In general, the nth derivative y<">, or /^"^(a;), is the deriva- tive of the derivative of order (n — 1). If, in thus forming the successive derivatives, we never obtain a function which has no derivative, we may imagine the process carried on indefinitely. In this way we obtain an unlimited sequence of derivatives of the func- tion f(x) with which we started. Such is the case for all functions which have found any considerable application up to the present time. The above notation is due to Lagrange. The notation /)^y, or D^f(x)f due to Cauchy, is also used occasionally to represent the 7ith derivative. Leibniz' notation will be given presently. 7. RoUe's theorem. The use of derivatives in the study of equa- tions depends upon the following proposition, which is known as RoUe's Theorem: Let a and b be two roots of the equation f{x)= 0. If the function f(x) is continuous and possesses a derivative in the irUerval {ay ft), the equation f'(x) = has at least one root which lies between aandb. For the function f{x) vanishes, by hypothesis, for x = a and x = b. If it vanishes at every point of the interval (a, i), its derivative also vanishes at every point of the interval, and the theorem is evidently fulfilled. If the function /(a;) does not vanish throughout the inter- val, it will assume either positive or negative values at some points. Suppose, for instance, that it has positive values. Then it will have a maximum value M for some value of x, say Xj, which lies betweei: a and b (§ 3, Theorem II). The ratio f{x, + h)-f{x,) h ' DKBIVATIVES AND DIFFERENTIALS [I. §8 k is taken positive, is necessarily negative or else zero. Uia limit of this ratio, i.e. /'(xi), cannot be positive; i.e. /*(«!) S®- ^^ ^ ^® consider /'(a:i) as the limit of the ratio /(x,-h)-f(x,) — h where A U poeitive, it follows in the same manner that f'(xi) > 0. two results it is evident that/'(xi) = 0. 8. Lew of the mean. It is now easy to deduce from the above theorem the important law of the mean:'**' Lei f{x) be a eontinuotu function which has a derivative in the imierval (a, 6). Then a) m-f{a) = (h-a)f(c), where e %» a number between a and h. In order to prove this formula, let <^ (x) be another function which hea the same properties as f{x)) i.e. it is continuous and possesses a derivative in the interval (a, b). Let us determine three constants, A,BfCj such that the auxiliary function ^(x)=^Af(x) + B<f>(x)-\-C TMiishes for x = a and for x = b. The necessary and sufficient conditions for this are i4/(a)+ 5^(«)+ C = 0, Af(b) + B<t»(b)+ C = 0; and these are satisfied if we set ^ =^(a)-^.(*)» ^=/W-/(a), C=f(a)<l>(b)-f(b)<f>(ay The new function ^(x) thus defined is continuous and has a derivative In the interval (a, b). The derivative i/^'(ar) = A f'(x) + B <f>'(x) there- fore ranishes for some value c which lies between a and b, whence, replaoiag A and B by their values, we find a relation of the form It is merely neceesary to take it»(x) = x in order to obtain the equality which was to be proved. It is to be noticed that this demonstration does not presuppose the continuity of the derivative /'(a;). .. The French also use "Formule de la My«MM M s qrBMym. Othar BngUah lynonymi are •• Average value theorem " ••e lUM valiM iheowD." — Tkani. I, §8] FUNCTIONS OF A SINGLE VARIABLE 9 From the tiieoieiii just proven it follows tliat it the derivative f'(x) is zero at each point of the interval (a, b), the function f{x) has the same value at every point of the interval ; for the applicar tion of the formula to two values Xiy x,, belonging to the interval (a, b)f gives /(x,)=/(xj). Hence, if two functions have the same derivative, their difference is a constant; and the converse is evi- dently true also. If a function F(x) be given whose derivative is f{x), all other functions which have the same derivative are found by adding to F{x) an arbitrary cotistant.* The geometrical interpretation of the equation (1) is very simple. Let us draw the curve A MB which represents the function g =f(^x) in the interval (a, b). Then the ratio [/(*) —/(«)]/(* — «) is the slope of the chord AB, while f'(c) is the slope of the tangent at a point C of the curve whose abscissa is c. Hence the equation (1) expresses the fact that there exists a point C on the curve A MB, between A and B, where the tangent is parallel to the chord AB. If the derivative f\x) is continuous, and if we let a and b approach the same limit Xq according to any law whatever, the number c, which lies between a and b, also approaches Xo> a-nd the equation (1) shows that the limit of the ratio m-m b — a is /'(^o)- The geometrical interpretation is as follows. Let us consider upon the curve y —f{x) a point M whose abscissa is Xq, and two points A and B whose abscissae are a and b, respectively. The ratio [f(b)—f{a)']/{b — a) is equal to the slope of the chord AB, while f'(Xo) is the slope of the tangent at ^f. Hence, when the two points .4 and B approach the point M according to any law whatever, the secant yl 5 approaches, as its limiting position, the tangent at the point M. • This theorem is sometimes applied without due regard to the conditions imposed in its statement. I^t/(x) and <p{x), for example, be two continuous functions which have derivatives /'(x), <f>'(x) in an interval (a, 6). If the relation /'(ar) <p{x)-/{r) 4>'{x) = is satisfie*! by these four functions, it is sometimes accepted as proved that the deriva- tive of the function// 0, or [/'(x) <t> (x) - f{x) 4>'{x)] / ^«, is zero, and that accordingly //0 is constant in the interval (a, 6). But this conclusion is not absolutely rigorous unless tlje function {x) does not vanish in the interval (a, 6). SupiKwe, for instance, that {x) and 4>'(x) b»th vanish for a value c between a and b. A function /(jr) equal to Cx4>{x) between a and c, and to C^tpix) In'twcen c and 6. where C^ and Cj are dif- ferent constants, is continuous and has a derivative in the interval (a, 6), and we have f'(x) <p (x) -f{x) (p'(x) - for every value of x in the interval. The geometricftl Interpretation is apparent. 10 DERIVATIVES AND DIFFERENTIALS [I, §9 This does not hold in general , however, if the derivative is not ooDtuiuous. For instance, if two points be taken on the curve y s x', on opposite sides of the y axis, it is evident from a figure that the direction of the secant joining them can be made to approach any arbitrarily assigned limiting value by causing the two points to approach the origin according to a suitably chosen law. The eqtiation (T) is sometimes called the generalized law of the SMon. From it de I'Hospital's theorem on indeterminate forms fol- lows at once. For, suppose f{a) — and ^ (a) = 0. Replacing h by Jr in (lOf we find where j'j lies between a and x. This equation shows that if the ratio f'(x)/<^'(x) approaches a limit as x approaches a, the ratio f{x)/^{x) approaches the same limit j if f(a) = and <f>(a) = 0. 9. Generalizations of the law of the mean. Various generalizations of the law of the mean have been suggested. The following one is due to Stieltjes {Bulletin de la SotAiti Math^matigue, Vol. XVI, p. 100). For the sake of definiteness con- ■ider three functions, /(x), g{x), h{x), each of which has derivatives of the first and aecond orders. I^et a, b, c be three particular values of the variable (a < 6 < c). Lei .^ be a number defined by the equation and let *(«) = /(a) g{a) h{a) f{h) g{b) h{b) /(c) g{c) h{c) f{o) g{a) h{a) /(&) g{b) h{b) fix) 9{x) h{x) 1 a a2 -A 1 6 62 1 c c2 1 a a2 -A 1 6 62 1 X X2 0, be aa auxiliary function. Shice this function vanishes when a; = 6 and when X = c, iu derivaUve must vanish for some value f between 6 and c. Hence /(a) 9 {a) h{a) /(&) 9{b) h{b) rU) fl^(f) A'(i-) -A 1 a o2 1 6 62 1 2t = 0. If 6 be replaeed by z In the left-hand side of this equation, we obtain a function rll******* ▼antahea when x = a and when x = 6. Its derivative therefore van- lahee for some value of x between a and 6, which we shall call {. The new •qaatlon thus obcained it /(a) /'(O fV) 9 (a) 9'(C) h{a) hV) -A a2 2r = 0. FtoaMy, repUelng f hy x In the lefuhand side of this equation, we obtain a func- lion of X which vanlabee when x = e and when x = f . Its derivative vanishes I, §10] FUNCTIONS OF SEVEUAL VARIABLES 11 for some value 17, which lien between ( and ^ and therefore between a and c Hence A muBt have the value -n /(a) g{a) h (a) r{i) 9'{i) h'{k) riyi) ff"(v) fi"iri) where { lies between a and 6, and »j lies between a and c. This proof does not presuppone the continuity of the second deriratbei /"(x), g"{x), h"{x). If these derivatives are continuous, ami if the values a^b^e approach tlie same limit Xq, we have, in the limit, lim^=i / (Xo) g (Xo) h (Xo) r{Xo) g'(Xo) A'(xo) /"(Xo) g^'ixo) /i"(xo) Analogous expressions exist for n functions and the proof follows the same lines. If only two functions /(x) and g (x) are taken, the formulae reduce to the law of the mean if we set g{x) = I. An analogous generalization has been given by Schwarz (Annali di Math&- matica, 2d series, Vol. X). II. FUNCTIONS OF SEVERAL VARIABLES 10. Introduction. A variable quantity <i> whose value depends on the values of several other variables, x, y, z, •••, t, which are in- dependent of each other, is called a function of the independ- ent variables x, y, «, •••, t; and this relation is denoted by writing o) =/(a;, i/,z,--y t). For definiteness, let us suppose that o> =f(x, y) is a function of the two independent variables x and y. If we think of X and y as the Cartesian coordinates of a point in the plane, each pair of values (x, y) determines a point of the plane, and con- versely. If to each point of a certain region A in the xy plane, bounded by one or more contours of any form whatever, there corresponds a value of w, the function /(x, y) is said to be defined in the region A. Let (xo, yo) be the coordinates of a point Mq lying in this r» . The function f{x, y) is said to be continuous for the pair of r (*o» yo) ifi corresponding to any preassigned positive number c, another positive number tf exists such that |/(xo -f A, yo 4- k) — /(Xo, yo)| < < whenever \h\< rj and \k\< rj. This definition of continuity may be interpreted as follows. Let us suppose constructed in the xy plane a square of side 2i; about 3/0 as center, with its sides parallel to the axes. The point AT. 12 DERIVATIVES AND DIFFERENTIALS [I, §11 whoM oooidinates are »o 4- *, yo + *, will lie inside this square, if |A|< If and |A| < ij. To say that the function is continuous for the pair of Tidues (*•, yo) amounts to saying that by taking this square ■affioieoUy smaU we can make the difference between the value of the function at M^ and its value at any other point of the square less than c in absolute value. It is evident that we may replace the square by a circle about (x^ y,) as center. For, if the above condition is satisfied for all pointo inside a square, it will evidently be satisfied for all points inside the inscribed circle. And, conversely, if the condition is satisfied for all points inside a circle, it will also be satisfied for all points inside the square inscribed in that circle. We might then define conti nuity by saying that an rj exists for every c, such that whenever VA* + k* < 17 we also have \f(xo -h A, yo + k) -f(xo, yo) | < c The definition of continuity for a function of 3, 4, • • • , n inde- pendent variables is similar to the above. It is clear that any continuous function of the two independent Tariables x and y is a continuous function of each of the vai-iables taken separately. However, the converse does not always hold.* IL Partial derivatives. If any constant value whatever be substi- tuted for y, for example, in a continuous function /(ic, y), there results a continuous function of the single variable x. The deriva- tive of this function of x, if it exists, is denoted by f^(x, y) or by a>^. Likewise the symbol w^, or /^ (x, y), is used to denote the derivative of the function /(oj, y) when x is regarded as constant and y as the independent variable. The functions f^ (x, y) and f^ (x, y) are called the partial derivatives of the function /(x, y). They are themselves, in general, functions of the two variables x and y. If we form their partial derivatives in turn, we get the partial derivatives of the sec- ond order of the given function /(x, y). Thus there are four partial deriTatives of the second order,/^(x, y),/^(x, y),/,^(x, y), /^(.x, y). The partial derivatives of the third, fourth, and higher orders are ft for IBftaBM, the function /(z, y), which is equal to 2 xy / (x^ + y^ when I two TsHablM c and y ar« not hoth zero, and which is zero when x — y = 0. It is It It ■ ooDtinuoua function of x when y is constant, and trice versa. MttMtailMM h It not a cootinuoua function of the two independent variables x and y lor tiM pair of vahMt s s 0, y = 0. For, if the point (x, y) approaches the origin upon Ikt llao a V y, tbt fnaoUoo/Cz, y) approached the limit 1 , and not zero. Such functions tevo ktoa aladlod by Batra la hit thetli. I, §iij FUNCTIONS OF SEVERAL VARIABLES 13 defined similarly. In general, given a function a> =/(ar, y, «, ••, ^) of any number of independent variables, a partial derivative of the nth order is the result of n successive differentiations of the function /, in a certain order, with respect to any of the variables which occur in/. We will now show that the result doesrfiot depend upon the order in which the differentiations are carried out. Let us first prove the following lenA|a : Let o> =f(x, y) be a function of the two variqhles^Land y. Then fxy — fyxi provided that these two derivatives are contu To prove this let us first write the expression U =/(x 4- Ax, y + Ay) -f(x, y -f Ay) -f(x -f Ax, y) hV(x, y) in two different forms, where we suppose that x, y, Ax, ^^ have definite values. Let us introduce the auxiliary function <^00=/(« + Ax, i;)-/(x, v), where i^ is an auxiliary variable. Then we may write u = <f>(y + ^y)-<f>(y)- Applying the law of the mean to the function <t>{v)f we have U = Ay<t>,(y + ^Ay), where < d < 1 ; or, replacing <f>^ by its value, U = Ay[/,(x -h Ax, y 4- ^Ay) -/,(x, y -f ^Ay)]. If we now apply the law of the mean to the function /^ (m, y + dAy), regarding u as the independent variable, we find f/ = Ax Ay/^^(x -f d' Ax, y + ^Ay), < ^' < 1. From the symmetry of the expression U in x, y. Ax, Ay, we see that we would also have, interchanging x and y, f/ = Ay Ax/^ (X + $[ Ax, y + $^ Ay), where ^, and 0[ are again positive constants less than unity. Equat- ing these two values of U and dividing by Ax Ay, we have /^(x -I- ^, Ax, y 4- d, Ay) =^(x 4- ^'Ax, y 4- ^Ay). Since the derivatives /^(x, y) and/^(x, y) are supposed continuous, the two members of the above equation approach f^ (x, y) and fwxi^f y)> respectively, as Ax and Ay approach zero, and we obtain the theorem which we wished to prove. 14 DERIVATIVES AND DIFFERENTIALS [I, §11 It is to be noticed in the above demonstration that no hypothesis whatever is made concerning the other derivatives of the second order, /^ and /^ The proof applies also to the case where the function /(x, y) depends upon any number of other independent variables besides x and y, since these other variables would merely have to be regarded as constants in the preceding developments. Let us now consider a function of any number of independent ▼enables, M ^ ^, .. and let O be a partial derivative of order n of this function. Ajiy permutation in the order of the differentiations which leads to O can be eCFected by a series of interchanges between two successive differentlbttions ; and, since these interchanges do not alter the resul^is we have just seen, the same will be true of the permuta- tioi^Hsidered. It follows that in order to have a notation which is dw ambiguous for the partial derivatives of the wth order, it is sufficient to indicate the number of differentiations performed with respect to each of the independent variables. For instance, any wth derivative of a function of three variables, w =/(aJ, y, «), will be represented by one or the other of the notations where p -\- q -{■ r = n.* Either of these notations represents the result of differentiating / successively p times with respect to x, q times with respect to y, and r times with respect to z, these oper- ations being carried out in any order whatever. There are three distinct derivatives of the first order, /^, /y, /^ ; six of the second order, /^, /^, /^, /^, /^, /^ ; and so on. In general, a function of p independent variables has just as many distinct derivatives of order n as there are distinct terms in a homo- geoeous polynomial of order niup independent variables ; that is, (n + l)(n-f 2).(n-f-jg-l) 1.2....(^-2)(^-l) ' as is shown in the theory of combinations. Praetieal ruUi. A certain number of practical rules for the cal- oulatioD of derivatiyes are usually derived in elementary books on •Tb» MUtton A,^,r(x, y, t) is OMd instead of the notation /j^J^r (x, y, z) for ■Inplleitj. Tbna the iM>Ution/^(x, y), used in place of /;;(«, y), is simpler and •qoall J ekar. — TaAM s. 1, §11] FUNCTIONS OF SKVERAL VARIABLES Ifi the (Calculus. A table of such rules is appended, the function and its derivative being placed on the saiae line : y = a'y y' = a' log a, where the symbol log denotes the natural logarithm ; y = logx, ^-i^ y = sin Xj y' = cos a- ; y = cos a?, y' = — sin X ; «# *M» Q 1*/% ain o^ u'- 1 : y = arc oiu X| ^ iVnriT.' y = arc tan ar, «'- 1 ; ^-l+x»' y = wv, y = u'v 4- uv' ; y =/(«), y.=f(y)u,', y =/(«. ", «>), y. = n,f,-^v,f,^w,f^ The last two rules enable us to find the derivative of a function of a function and that of a composite function if /,,/»»/» are con- tinuous. Hence we can find the successive derivatives of the func- tions studied in elementary mathematics, — polynomials, rational and irrational functions, exponential and logarithmic functions, trigonometric functions and their inverses, and the functions deriv- able from all of these by combination. For functions of several variables there exist cfitain formulae analogous to the law of the mean. Let us consider, for definite- ness, a function /(x, y) of the two independent variables x and y. The difference /(x + ^, y 4- k) —f{x, y) may be written in the form /(x 4- h, y 4- k) -/(x, y) = [/(x + ^, y -f k) -f(r, y -\- A-)] + [/(^, y4-A:)-/(^>y)]» to each part of which we may apply the law of the mean. We thus find f(x -^h,y + k) -/(x, y) = hf,(x + eh,y-^k)-\- kf^(x, y 4- B'k), where 6 and $* each lie between zero and unity. This formula holds whether the derivatives/, and/^ are continu- ous or not. If these derivatives are continuous, another formula, Uki 16 DERIVATIVES AND DIFFERENTIALS [I, §12 timiUr to the above, but involving only one undetermined number 0^ maj be employed.* In order to derive this second formula, con- sider the auxiliary function <ft(t) =f(x + ht, y -\- kt), where x, y, A, and k have determinate values and t denotes an auxiliary variable. Applying the law of the mean to this fjtnction, we find ^(1)-<^(0)=<^W, b<^<i. Now ^{i) is a composite fundtion of t^ and its derivative <^'(^) is equal to A/,(x + V, y 4- A;^ + ^fy{x 4- ht, y + kt)-, hence the pre- ceding formula may be written in the form f(x-k-h,y-{'k)'-f{x,y) = hf,{x-iteh,y + ek) + kf^{x + eh,y+ek), 13. Tangent plane to a surface. We have seen that the derivative of a function of a single variable gives the tangent to a plane curve. Similarly, the partial derivatives of a function of two variables occur in the determination of the tangent plane to a surface. Let (2) z = F(x, y) be the equation of a surface 5, and suppose that the function F(x, y), together with its first partial derivatives, is continuous at a point («o» yo) of the xy plane. Let Zq be the corresponding value of «, and il/o (a^o, yo, «o) the corresponding point on the surface S. If the equations (3) x=f{t), y = <t>(t), z = tf;(t) reprasent a curve C on the surface S through the point Mq, the three functions /(<), <t>(t), ^(^), which we shall suppose continuous and differentiable, must reduce to Xq, yo, Zq, respectively, for some ralue t^ of the parameter t. The tangent to this curve at the point J/« is given by the equations (§ 5) ^ f{to) <!>%) ^'(to) ' Since the curve C lies on the surface S, the equation ^(t)=F[f(t), <f>(t)'\ must hold for all values of ^; that is, this relation must be an identity • A a o tlfr lonDolA may be obtained which involves only one undetermined number d, Bad which holds even whan thederlvatives/e and/^ are discontinuous. For the applica- Uoiiof th« Uw of the mean to the auxiliaiy function 0(0 =/(x + H y + *)+/(«, y + AO ^ ^a)-^(O) = 0'(^, 0<tf<l, /(« + *, If + *) -/(«» y) = A/,(x + tfA, y + t) + k/y(z, y + ek), 0<d<\. The op eiat loM performed, and hence the final formula, all hold provided the deriva- HvM/^Md/ymerely exist at the pointo (x + A<. y + *), (x, y + AO, 0<<^1. — Trans. I, § 13] FUNCTIONS OF SEVERAL VARIABLES 17 in t. Taking the derivative of the second member by the rule for the derivative of a composite function, and setting t = <©, we have (5) fC.) =/'(<•) ^'.. + *'W^v We can now eliminate f{t^, ♦'('•)' ^'(M between the equations (4) and (5), and the result of this elimination is (6) Z-x, = {X-x,)F^+(^Y-y,)F,^. This is the ecjuation of a plane which is the locus of the tangents to all curves on the surface through the point Mq. It is called the tan- gent plane to the surface, 18. Passage from increments to derivatives. We have defined the BUCcesBive derivatives in terms of each other, the derivatives of order n being derived from those of order (/i - 1), and so forth. It is natural to inquire whether we may not define a derivative of any order as the limit of a certain ratio directly, with- out the intervention of derivatives of lower order. We have already done some- thing of this kind lor f^y (§ 11); for the demonstration given above shows that/a^ is the limit of the ratio /(x -h Aa, y -f Ay) -/(x + Aa;, y) -/(g, y + Ay) -f /(g, y) Ax Ay as Ax and Ay both approach zero. It can be shown in like manner that the second derivative /" of a function /(x) of a single variable is the limit of the ratio fix -t- Ai + ^) -/(x -i- ^0 -/(x 4- h^) -H/(x) hxho as h\ and hi both approach zero. For, let us set /i(«)=/(x + Ai)-/(x), and then write the above ratio in the form /i(x-f A»)-/i(x) ^ /{(x-f-gA,) o«^<l; hxh^ hi or hi The limit of this ratio is therefore the second derivative /", provided that derivative is continuous. Passing now to the general case, let us consider, for definiteness, a function of three independent variables, w =/(x, y, z). Let us set Ajw =/(x + A, y, z) -/(x, y, «), aJw =/(x, y + *, «) -/(x, y, «), a[« =/(x, y, z + -/(x, y, 2), where aJ w, aJ w, A^ w are i\\Q first increnxents of w. If we consider A, Jr, < as givW) constants, then the.se three first increments are themselves functions of x, y, s, and we may form the relative increments of these functions corresponding to 18 DERIVATIVES AND DIFFERENTIALS [I, §13 tDcremenu Ai, i^i, h of the variables. This gives us the second increments, ^^^, A*i A* w, • • • . This process can be continued indefinitely ; an increment of Older n would be defined as a first increment of an increment of order (n — 1). 8iiieo we may invert the order of any two of these operations, it will be suffi- doDt to tndifiatt the successive increments given to each of the variables. An increment of order n would be indicated by some such notation as the following : AC)« = aJ« A^^ . . . Ai' aJ' . . . aJ» aJ . . . a';/{z, y, «), p + q -^ r = n^ and where the increments h, k, I may be either equal or This increment may be expressed in terms of a partial derivative of order », being equal to the product X /.f»i,r(x + $ihi + • • • + 0php, y + eikx + . . . + e'^kq, z 4- e^h + • • • + K'lr), where every B lies between and 1. This formula has already been proved foi first and for second increments. In order to prove it in general, let us assume that it holds for an increment of order (n — 1), and let 0(2, y, z) = aJ« . . . aJ'' Ajt . . . aJ'A^ . . . aIv. Then, by hypothesis, #(Xt y, «) = ^- • • Ap*l • • Kh • • • lrfxP-l^zr{X-\-e2h^-\-' . . +^p^, y + . . ., Z+ . . .). Bat the nth increment considered is equal to 0(x + ^i, y, z) — 0(x, y, z) ; and if we apply the law of the mean to this increment, we finally obtain the formula sought. Conversely, the partial derivative fxP^flz* is the limit of the ratio A^AJi'-. ■ A^A^t. .. A^^A^ • • A,V h\hi' hpkiki- • -kqli" Ir M all the increments A, k, I approach zero. It is interesting to notice that this definition is sometimes more general than the Qsoal definition. Suppose, for example, that w =/(x, y) = <P{x) + yf/ (y) is a function of x and y, where neither <f> nor ^ has a derivative. Then <a also has no first derivative, and consequently second derivatives are out of the question. In the ordinary sense. Nevertheless, if we adopt the new definition, the deriva- UTe/ay is the limit of the fraction fix -\.h,y-\-k) -fix + h, y) -fix, y + k) -\-f{z, y) hk irhich Is equal to ^(x -f ») -f f (y -t- ie) - 0(g -f. A) - ^(y) ~ »(x) -- ^(y + fe) -f ^(x) -\- ^(y) hk But the numerator of this ratio is identically zero. Hence the ratio approaches wro as a limit, and we find /ry = 0.* * ▲ ilmUar remark may be made regarding functions of a single variable. For example, the ftuiotlon/(z) = z* cos 1/x has the derivative X z •J>d/'(«) has no derivative for a; = 0. But the ratio /(2a)-2/(a)-f/(0) -J or SaeotO/So) — aaeoeO/or), has the limit sero when a approaches sero. I, il4] THE DIFFERENTIAL NOTATION 19 in. THE DIFFERENTIAL NOTATION The differential notation, which has been in use longer than any other,* is due to Leibniz. Although it is by no means indispensable, it possesses certain advantages of symmetry and of generality which are convenient, especially in the study of functions of several varia- bles. This notation is founded upon the use of infinitesimals. 14. DifferentUls. Any varuible quantity which approaches zero as a limit is called an infinitely small quantity^ or simply an infinitesi- mal. The condition that the quantity be variable is essential, for a constant, however small, is not an infinitesimal unless it is zero. Ordinarily several quantities are considered which approach zero simultaneously. One of them is chosen as the standard of compari- son, and is called the principal infinitesimal. Let a be the principal infinitesimal, and ft another infinitesimal. Then $ is said to be an infinitesimal of higher order ivith respect to a, if the ratio ft/ a approaches zero with a. On the other hand, ft is called an infini- tesimal of the first order with respect to a, if the ratio ft/ a approaches a limit K different from zero as a approaches zero. In this case ^ = /iT + e, a where c is another infinitesimal with respect to a. Hence ft=a{K + t)= Ka-\- a€, and Ka is called the principal part of ft. The complementary term arc is ah infinitesimal of higher order with respect to a. In general, if we can find a positive power of a, say a*, such that ft/a^ approaches a finite limit K different from zero as a approaches zero, ft is called an infinitesimal of order n with respect to a. Then we have | = a: + ., or ft = c^{K + €)= ^o- -f a"c. The term KoT is again called the principal part of ft. Having given these definitions, let us consider a continuous func- tion y =f(x), which possesses a derivative f\x). Let Ai! be an * With the possible exception or Newton's notation. — T&AK& 20 DKRIVATIVES AND DIFFERENTIALS [I, §14 increment of x, and let Ay denote the corresponding increment of y. From the very definition of a derivative, we have where c approaches zero with Ax. If Ax be taken as the principal infinitesimal, Ay is itself an infinitesimal whose principal part is /*(x)Ax.* This principal part is called the differential of y and is denoted by dy. dy=f{x)^. Whan /(x) reduces to x itself, the above formula becomes dx = Ax'y and hence we shall write, for symmetry, dy=f'(x)dx, where the increment dx of the independent variable x is to be given the same fixed value, which is otherwise arbitrary and of course variable, for all of the several dependent functions of x which may be under consid- eration at the same time. Let us take a curve C whose equation is y=f(^)f and consider two points on it, M and M', whose abscissae are x and x + dx, respectively. In the triangle MTN we have NT = MN tan Z TMN = dxf'(x). Fio. 1 " \ / Hence NT represents the differential dy, while Ay is equal to NM'. It is evident from the figure that M'T if an infinitesimal of higher order, in general, with respect to NT, as ir approjMjhes A/, unless MT is parallel to the x axis. Successive differentials may be defined, as were successive deriv- ati768, each in terms of the preceding. Thus we call the differ- ential of the differential of the first order the differential of the second order, whore dx is given the same value in both cases, as abore. It is denoted hy d^y. d*y = d(dy) = \_fXx) dx-] dx =/"(x) (dx)\ SimUarly, the third differential is rf»y = d{d^y) = [/"(a;) rfx«] dx =f"(x) (dxy, r > T a ^ ^y y d J » < } ^ i^*!!iJ!L2!r^'l7! ^u^^ **•"' "'^"'** ***' "»«« where/'(a;) = 0. It Is, how- 7^S^Ti.JTl^Ji"r?' ''^"'''"" "' ^1/ =/'(.) Ax in this case also. •fm laoogb it U not the prinolpAl (wrt of Ay.— Traks. I, §14] THE DIFFKKKNTIAL NOTATION 21 and so on. In general, the di£ferential of the differential of ordex The derivatives /'(x), /"(x), • •, /^"^x), • • • can be expressed, on the other hand, in terms of differentials, and we have a new nota- tion for the derivatives : To each of the rules for the calculation of a derivative corresponds d rule for the calculation of a differential. For example, we have c? x*" = mx^'^dXf da* = a* log a dx ; d log X = — > d sin x = cos xdx , • • • ; X , . dx 7 i. ^ cf arc sin 2 = > a arc tan x = Let us consider for a moment the case of a function of a function. Let y =f(u), where u is a function of the independent vaiiable x. Then ^. , whence, multiplying both sides by rfx, we get yx^=/Wx ?/,'/x; that is, </y =f(ii)du. The formula for rfy is therefore the same as if u were the inde- pendent variable. This is one of the advantages of the differential notation. In the derivative notation there are two distinct formulae, to represent the derivative of y with respect to x, according as y is given directly as a function of x or is given as a function of x by means of an auxiliary function u. In the differential notation the same formula applies in each case.* li y = f(u, V, xv) is a composite function, we have at least if /^,/,,/^ are continuous, or, multiplying by <ix, y^dx = u^dxf, + v^dxf^ + tr,<&/^ ; • This particular advantajfe Is slight, however ; for the last formula above is equally well a general one and covers both the cases mentioned. — Trans. vdu — udv S2 DERIVATIVES AND DIFFERENTIALS [I, §16 ihit 18, dy =fndu -\-f,dv -^f^dw. Thus we have, for example, d{uv)=udv -\-vdUf rff-j = The same rules enable us to calculate the successive differentials. Let us aeek to calculate the successive differentials of a function y =/(«), for instance. We have already dy =f\u)du. In order to calculate <?y, it must be noted that du cannot be regarded as fixed, since u is not the independent variable. We must then calculate the differential of the composite function /'(w) (iw, where u and du are the auxiliary functions. We thus find d^y =f\u)du^ +fiu)d^u. To calculate rf'y, we must consider «Py as a composite function, with ti, du, d^u 2i8 auxiliary functions, which leads to the expression d*y =f"(u) dv* + 3/"(w) dud^u +f{u) d^u ; and so on. It should be noticed that these formulae for d^y^ d^y, etc., are not the same as if w were the independent variable, on account of the terms d^u, d^Uy etc.* A similar notation is used for the partial derivatives of a function of several variables. Thus the partial derivative of order n of f(Xf y, «), which is represented by fxP^^r in our previous notation, u represented by s^Ms^' -? + ? + '• = "' in the differential notation. t This notation is purely symbolic, and in no sense represents a quotient, as it does in the case of functions of a single variable. 15. Toul differentials. Let u =/(a;, y, «) be a function of the three independent variables x, y, z. The expression * This diMdvanUfftt would M«m completely to offset the advantage mentioned •bore. BUiotly apeaklng, we should distinguish between dPj/ and dly, etc. — Tran«). t This Oit of the letter d to denote the partial derivatives of a function of several VMtehUi Is dos to JaooM. Before his time the same letter d was used as is used for Uwi d«rivativMi of a function nf h Niti^le variable. 1,516] THE DIFFKKENTIAL NOTATION 28 is called the total differential of w, where rfx, rfy, dz are three fixed increments, which are otherwise arbitrary, assigned to the three independent variables x, y, z. The three products ^^' ^^^' a;^* are called partial differentials. The total ditferential of the second order d^utis the total differ- ential of the total differential of the first order, the increments dxy dy, dz remaining the same as we pass from one differential to the next higher. Hence or, expanding, / c^f av av \ rf«a>= -Idx + ^dy + -^dz)dx \ cx^ ex cy cxcz I -I- 2 r^ dxdy-^2 -^ dxdz-^2 ^ dy dz. ex cy "^ ox cz Cy cz If d^f be replaced by df^y the right-hand side of this equation becomes the square of We may then write, symbolically, it being agreed that df^ is to be replaced by d^f after expansion. In general, if we call the total differential of the total differential of order (71 —1) the total differential of order n, and denote it by <i"w, we may write, in the same symbolism. \Cx Cy ^ dz J where a/" is to be replaced by d^f after expansion ; that is, in out ordinary notation, ti DERIVATIVES AND DIFFERENTIALS [I, §15 where \b the coefficient of the term a^b'fif in the development of (a + J H-c)". For, suppose this formula holds for d" w. We will show that it then holds for <f*"^*«; and this will prove it in general, since we have Already proved it for n = 2. From the definition^ we find whence, replacing ^+y by ^/"^S the right-hand side becomes Hence, using the same symbolism, we may write Note, Let us suppose that the expression for G?a>, obtained in any way whatever, is (7) dia = Pdx-h Qdy + Rdz, where P, Q, R are any functions x, y, «. Since by definition we must have where dxy dy, dz are any constants. Hence The tingle equation (7) is therefore equivalent to the three separate aquatuma (8)i and it determineH all three partial derivatives at once. 1,516] THK DIFFERENTIAL NOTATION 25 In general, if the nth total differential be obtained in any way whatever, . _,^ j „j ^ , . then the coefficients C^^ are respectively equal to the corresponding nth derivatives multiplied by certain numerical factors. Thus all these derivatives are determined at once. We shall have occasion to use these facts presently. 16. Successive differentials of composite funaions. Let <d = F(u, v, w) Ix; a composite function, //, v, w being themselves functions of the independent variables a?, y, «, t. The partial derivatives may then be written down as follows : dx du dx dv dx dw dx do» du> dz __dFdu 'dFdv du dy dv dy _dFdu dFdv du dz dv dz , dFdw CW cy dFdw dw dz da> dt dFdu dFdv ~ du dt^ dv dt ,^dw dw dt If these four equations be multiplied by and added, the left-hand side becomes dx, dy, dz, dtf respectively, ^<i> , , dio . , dto . du) . — dx + -^ dy + -^ dz ■{- -^dt, dx dy ^ dz <^y that is, d(o J and the coefficients of dF dF du dv dF dw on the right-hand side are du, dv, dw, respectively. Hence dF dF dF (9) c?a) = T— du -f -— rfr -f ^— dw, ^ ^ CU CV CIV and ive see that the expression of the total differential of the first order of a composite function is the same as if the auxiliary function* ivere the independent variables. This is one of the main advantages of the differential notation. The equation (9) does not depend, in form, either upon the number or upon the choice of the independent variables ; and it is equivalent to as many separate equations as there are independent variables. To calculate d^u), let us apply the rule just found for dut, noting that the second membt»r of (9) involves the six auxiliary functions u, v, w, du, dr, die. We thus find 26 DERIVATIVES AND DIFFERENTIALS [I, §16 rf«« = -__. rf|4« 4- — — - du dv + r-TT- dudw-{--r- d*u • ^* dudv ducw du '^'r—r-dudw-k"r-r-dvdw+ |-t c?«;^ + ^ cPw, or, simplifying and using the same symbolism as above, d^^^f^^du-^-^dv^^dwj ^^d^u+^d^^^d^u:. This formula is somewhat complicated on account of the terms in </*u, d*Vy d^Wj which drop out when u, v, w are the independent variables. This limitation of the differential notation should be borne in mind, and the distinction between d^ta in the two cases carefully noted. To determine d^my we would apply the same rule to (^<o, noting that cT'o) depends upon the nine auxiliary fimctions tf, r, IT, du, du, dw, d^u,d^v, d^w\ and so forth. The general expres- sions for these differentials become more and more complicated ; </*•» is an integral function of du, dv, dw, d^u, • • •, d'^Ujd^v, d!^w, and the terms containing d:*u, d^v, d^w are •r- (Tw 4- -r- dPv + ^— d^w. ou cv ow If, in the expression for c?" w, u, v, w, du, dv, dw, • • • be replaced by their values in terms of the independent variables, d^m becomes an integral polynomial in dx, dy, dz, • • • whose coefficients are equal (cf. Note, § 15) to the partial derivatives of w of order n, multiplied by certain numerical factors. We thus obtain all these derivatives at once. Suppose, for example, that we wished to calculate the first aqd second derivatives of a composite function <u=/(z*), where w is a function of two independent variables w = <^ (x, y). If we calculate these derivatives separately, we find for the two partial derivatives of the first order d(i> dutdu dto dto du dx du dx dy du dy ao) Again, taking the derivatives of these two equations with respect to X, and then with respect to y, we find only the three following distinct equations, which give the second derivatives: Kun THE DIFFERENTIAL NOTATION 27 (11) dx* " du*\dx) du d*i d*ot du du dm d^io _ d*ia/du [ 'd^ '"d^^\d^ )* ,dm d^ "*■ du d/ The second of these equations is obtained by differentiating the first of equations (10) with respect to y, or the second of them with respect to x. In the differential notation these five relations (10) and (11) may be written in the form (12) dm = du du, M 3*u> , , , dm j^ du' du If du and d*u in these formulae be replaced by du dll ^dx-^-dy and d^ dx ldx'-^2 C^u a«i dxdy'^'^'^y^-d^-^^^ redpectively, the coefficients of dx and dy in the first give the first partial derivatives of w, while the coefficients of dx^, 2dxdyj and dy^ in the second give the second partial derivatives of m. 17. Differentials of a product. The formula for the total differential of order n of a composite function becomes considerably simpler in certain .special cases which often arise in practical applications. Thus, let us seek the differential of order n of the product of two functions w = mv. For the first values of n we have dii) = V du -{- u dr, d^w = vd^u -\- 2dudv -\- ud^v, •••; and, in general, it is evident from the law of formation that rf"<i) = vd^u H- Cidvd''-Ui -\- C^d^vdr-'u ^ \-u^Vj where C,, C„ • • are positive integers. It might be shown by alge- braic induction that these coefficients are equal to those of the expansion of (a -f- />)* ; but the same end may be reached by the following method, which is much more elegant, and which applies to many similar problems. Observing that Cj, C,, • • • do not depend upon the particular functions u and v employed, let us take the 88 DERIVATIVES AND DIFFERENTIALS [I, §17 special functions u = <?*, v = e», where ar and y are the two inde- pendent variables, and determine the coefficients for this case. We thus find du-e'dxy d^u = e'dx^, ••, dv = e^dy, d^v = e^dy^ • • • ; and the general formula, after division by e^+^ becomes {dx 4- dyy = dx* + C^dydx""-^ + C^dfdx''-'^ -\ -{■ d^. Since dx and dy are arbitrary, it follows that n ^ n(n-l) n(n -1) "-jn -p + V) ^«==r •" 1.2 ' "' ^"" 1.2...J9 and consequently the general formula may be written (n)dr(uv) = vdru+'^dvd^-Ui + '^^^^~^ ^ d^vd^-^'u + '-' + ud^v. This formula applies for any number of independent variables. In particular, if u and v are functions of a single variable x, we have, after division by rfa;*, the expression for the wth derivative of the product of two functions of a single variable. It is easy to prove in a similar manner formulae analogous to (13) for a product of any number of functions. Another special case in which the general formula reduces to a simpler form is that in which w, v, w are integral linear functions of the independent variables x, y^ z. u= ax-h by+ cz+f , v = a'x -\- b'y + c'z +/' , w = a"x + b"y + c"z -[-/", where the coefficients o, a', a", b, 6', • • • are constants. For then we have du= adx -{- bdy -\' cdzy dv = a'dx -f b'dy -f- c'dz, dw = a"dx-^b"dy + c"dz, and all the differentials of higher order d^u, d^Vy dTw, where n> 1, ranish. Hence the formula for <i"ca is the same as if w, v, w were the independent variables ; that is, I, J 18] THE DIFFERENTIAL NOTATION 29 . (dF , dF . dF , Y> (irui = \-;r-du-\--r-dv-\--r- aw I . \rit cv cw I We proceed to apply this remark. 18. Homogeneous functions. A function ^(x, y, «) is said to be homogeneous of degree m, if the equation is identically satisfied when we set u — tx^ V =.tiiy w = tz. Let us equate the differentials of order 7i of the two sides of this equation with respect to t, noting that u, v, w are linear in f, and that du = X dty dv = y dty dw = « dt. The remark just made shows that y'^'^'^dv'^^^J = m(m -1) . .. (m - n -hl)r-^(x, y, z). If we now set / = 1, w, v, w; reduce to x, y, «, and any term of the development of the first member, becomes '^dx^di/^dz'-'^''^''^' whence we may write, symbolically, y'di'^'^d^'^''^) "^ "'('^ - 1) • • • (m - n+l)<t>{x, y, z), which reduces, for n = 1, to the well-known formula . , X ^<l> ^4> ^4> Various notations. We have then, altogether, three systems of nota- tion for the partial derivatives of a function of several variables, — that of Leibniz, that of Lagrange, and that of Cauchy. Each of these is somewhat inconveniently long, especially in a complicated calculation. For this reason various shorter notations have been devised. Among these one first used by Monge for the first and 80 DERIVATIVES AND DIFFERENTIALS [I, §19 •eoond derivatives of a function of two variables is now in common use. If « be the function of the two variables x and y, we set d% dz d^z _ d^z ._?!f. ^"^' ^"a^' '■"a^^' '"dxdy ^"dy^' md the total differentials dz and cPz are given by the formulae dz= pdx-\-qdi/, d^z = rdx^ -\- 2 s dx dy -\- 1 dy^. Another notation which is now coming into general use is the following. Let « be a function of any number of independent vari- ables Xi, Xj, x», •••,«» J then the notation ^'^"*"'"*^ dx'^dxt^'-dx'^ is used, where some of the indices ^i, a^, • • •, a„ may be zeros. 19. Applications. Let y =f{x) be the equation of a plane curve C with reqiect to a set of rectangular axes. The equation of the tangent at a point Mix, V) is Y-y = y'{X-x). The dope of the normal, which is perpendicular to the tangent at the point of tangency, is — l/y' ; and the equation of the normal is, therefore, {T-y)y'+{X-x)=:0. Let P be the foot of the ordinate of the point 3f, and let T and N be the points of intersection of the x axis with the tangent and the normal, respectively. The distance PN is called the subnormal ; Pr, the subtangent; MN, the normal; and MT, the tangent. From the equation of the normal the ab- scissa of the point iV is x + yy\ whence the subnormal is ± yy\ If we agree to call the length PN the subnormal, and to attach the sign + or the sign — according as the direc- tion PN is positive or negative, the subnormal will always be yy' for any position of the curve C. Likewise the subtangent is — y/y\ The leogthe MN and MT are given by the triangles MPN and MPT: MNz^y/Mp^PN^^yVTTy^, MT = V5p« + pT* = ^VTT?^, .-T* ***** pwN«ni may be given regarding these lines. Let us find, for iMlftDet, an the eorvee for which the subnormal is constant and equal to a given ombtr a. This amounta to finding all the functions y =/(x) which satisfy the eqoaUoa loT = a. The left-hand side U the derivative of yV2, while the I, Em.] EXKRCISES 81 right-hand side ia the derivative of ax. These functions can therefore diflsr only by a constant ; whence which is the equation of a parabola along the z axis. Ag|iin, if we seek the curves for which the subtangent is constant, we are led to write down the eqa»- tion ]/ /y — \/a\ whence X * logy = - + log C, or y = Ce^, a which is the e(}uaunn of a transcendental curve to which the x axis is an asymp- tote. To find the curves for which the normal is constant, we have the equation The first member is the derivative of — Va^ — y« ; hence - Vaa-y« = x + C, or (X + C7)9 + 1/2 = a2, which is the equation of a circle of radius a, whose center lies on the x axis. The curves for which the tangent is constant are transcendental curves, which we shall study later. Let y = /(x) and Y = F{x) be the equations of two curves C and C, and let Af, M' be the two points which correspond to the same value of x. In order that the two subnormals should have equal lengths it is necessary and suflficient that yr'=±yy'; that is, that Y^ = ± y^ i- C, where the double sign admits of the normals' being directed in like or in opposite senses. This relation is satisfied by the curves and also by the curves .« = >-<,^. r« = ^, a^ a^ which gives an easy construction for the normal to the ellipse and to the hyperbola. EXERCISES 1. Let p = f{d) be the equation of a plane curve in polar coordinates. Through the pole draw a line perpendicular to the radius vector OM, and let T and N be the points where this line cuts the tangent and the normal. Find expres- sions for the disunces OT, ON, MN, and MT in terms of /(<?) and f\e). Find the curves for which each of these distances, in turn, is constant. 2. Let y = /(x), t = (x) be the equations of a skew curve r, i.e. of a general space curve. Let N Fio. 3 S2 DERIVATIVES AND DIFFERENTIALS [I, Exs. to the point where the normal plane at a point 3f, that is, the plane perpendicu- lar to Um tangent at JT, meet« the z axis ; and let P be the foot of the perpen- dlcaUr from Jf to the « axis. Find the curves for which each of the distances PN and MN, in torn, is constant. [NoU. The« cures lie on paraboloids of revolution or on spheres.] S. IMennine an integral polynomial /(x) of the seventh degree in z, given that/(x) + 1 is divisible by (» - 1)* and f{x) - 1 by (x+1)*; Generalize the problem. 4. Show that if the two integral polynomials P and Q satisfy the relation vT^r^=QVl-x2, dP ndx Vl - pa Vl -x2 » is a positive integer. [ JTote. From the relation (a) l-P2=Q2(i_a;2) it follows that (b) -2PP'=Q[2Q'(l-x2)-2Qx]. The equation (a) shows that Q is prime to P ; and (b) shows that P' is divisible 5e. Let A (z) be a polynomial of the fourth degree whose roots are all dif- ferent, and let X = IT/ F be a rational function of t, such that ^' Q{t) ^' where Ri (0 is a polynomial of the fourth degree and P/ Q is a rational function. Show that the function U/ V satisfies a relation of the form dx kdt VR{x) VRiit) wbare l( is a constant [Jacobi.] INoU. Each root of the equation R(U/ V) = 0, since it cannot cause R'{x) to vanish, most cause UV - VU\ and hence also dx/db, to vanish.] ••. Show that the nth derivative of a function y = 0(m), where u is a func- tion of the independent variable x, may be written in the form W 1^ = ^i ♦'(«) + Y^ ♦"(") + ■• • + 7-^ «<")(»). wbart (b) da^ ' ' 1.2^ ' ' 1.2- d»tt» Ac^ d''u*-^ fc(Jk-l) jd»u*2* da^ l" dx» 1.2 ^ dx» "^ + (-l)»-tfcu*-i^ (fc=l,2....,n). [Ill* DOtlM that the nth derlvatiTe may be written in the form (a), where the oodBdenU At. A*. . A^ are independent of the form of the function <f>{u). I, Exb] exercises 83 To find their values, set ^(u) equal to u, u*, • • •, u" successively, and solve the resulting equations for ^i, At, • • • , A^. The result is the form (b).] 7*. Show that the nth derivative of <p{x^ is ^^^^^^ = (2x)-^(-)(x«) + n(n- l)(2x)"-»'-f"-"f -2, , . .. + "<" - ')•-(" -^P+ ») (2x)»-'.-».-..(x') + ■■■. 1 . 2' • 'P where p varies from zero to the last positive integer not greater than n/2, and whore 0<')(x'') denotes the tth derivative with re8pect to x. Apply this result to the functions e-**, arc sin x, arc tan x. 8*. If X = cos u, show that d«-»(l-x«)'"-* , ,, . ,1.3.6...(2m-l) , i = (- l)"'-! i i sin mu. dx"«-» m [OlIKDK RODRIUUEA.] 9. Show that Legendre's polynomial, ^ 1 d» 2 . 4 . 6 • • . 2 n dx" satisfies the differential equation Hence deduce the coefBcients of the polynomial. 10. Show that the four functions yi = sin (71 arc sin x), ys = sin (n arc cos x), ya = cos (n arc sin x), y^ = cos (n arc cos x), satisfy the differential equation (1 -x2)y"-xy' + n2y = 0. Hence deduce the developments of these functions when they reduce to poly- nomials. 11*. Prove the formula 1 d" 1 e* _(x.-.e.) = (-l).— . [Halphen.] 12. Every function of the form 2 = x ^ (y /x) + ^ (y /x) satisfies the equation rx« + 2 sxy + ty^ = 0, whatever be the functions and ^. 13. The function z = x0(x + y) + yt/'(x + y) satisfies the equation r-2a + t = 0, whatever be the functions </> and ^. S4 DERIVATIVES AND DIFFERENTIALS [I, Ei& 14. The function s = /[x + 4> (y)] aatisfies the equation pa = qr, whatever be the functions/ and ^. 15. The function « = x"^(y/x) + jr"^(y/«) satisfies the equation rx» + 2«cy + «y2 + px + gy = n^z, whaterer be the functions and ^. 16. Show that the function y = \x - ai\th{x) + \x - ai\<h{x) -\- '•• + \x - an\<f>n{x), where ^ (z), ^ (x), • • • , 0» (x), together with their derivatives, 0i (x), 03 (x), • • • , ^ (X), are continuous functions of x, has a derivative which is discontinuous forx = Oi, Os, •••, a.. 17. Find a relation between the first and second derivatives of the function t =/(Xi, u), where u = 0(xt, Xj); Xi, Xa, Xz being three independent variables, and /and two arbitrary functions. 18. Let/(x) be the derivative of an arbitrary function /(x). Show that 1 d^u _ 1 d^v udx^ ~ V dx2' where u = [f(x)]-i and » =/(x) [/'(x)]-i. ,19«. The nth derivative of a function of a function u = (y), where y = ^ (x), may be written in the form 1^0 = the sign of summation extends over all the positive integral solutions of the equation i + 2j + 3 A + • • • + iA: = n, and where p = i+j + "• + k. (Faa d» Bbumo, Quarterly Journal of Mathematics, Vol. I, p. 359.] CHAPTER II IBfPLICIT FUNCTIONS FUNCTIONAL DETERMINANTS CHANGE OF VARIABLE L IMPLICIT FUNCTIONS 20. A particular case. We frequently have to study functions for which no explicit expressions are known, but which are given by means of unsolved equations. Let us consider, for instance, an equation between the three variables ar, y, «, (1) F{x,y,z) = 0. This equation defines, under certain conditions which we are about to investigate, a function of the two independent variables x and y. We shall prove the following theorem : Let X — Xq, y = 1/oy ^ = ^o ^^ ct set of values which satisfy the eqtm- tion (1), and let us suppose that the function F, together with its first derivatives, is continuous in the neighborhood of this set of values.* If the derivative F, does not vanish for x = Xq, y = yoi z = Zq, there exists one and only one continuous function of the independent variables X and y which satisfies the equation (1), aiid which assumes the value Zq when X and y assume the values Xq and y^, respectively. The derivative F, not being zero for a; = a;©, y = yoj ^ = «o> let us suppose, for definiteness, that it is positive. Since F, F„ F^, F, are supposed continuous in the neighborhood, let us choose a positive number / so small that these four functions are continuous for all sets of values «, y, z which satisfy the relations (2) |a;-xo|</, \y-yo\^ly \z-z,\<l, and that, for these sets of values of aj, y, «, ^.(a^, y, «) > F, • In a recent article {Bulletin de la Soci^U Math^matique de France, Vol. XXXI, 1903, pp. 184-192) Groursat has showTi, by a method of successive approximations, that it is not necessary to malce any assumption whatever repardinjj F, and F^, even as to their existence. His proof makes no use of the existence of F, and f\. His general theorem and a sketch of his proof are given in a footnote to § 25. — Trans. 36 ge FUNCTIONAL RELATIONS [II.§20 whm P is some positive number. Let Q be another positive num- ber greiOer than the absolute values of the other two derivatives F-, F, in tlie same region. Giving X, y, « values which satisfy the relations (2), we may then write down the following identity : F(x,y,M)^F{x,,yo,z,)^F{x,y,z)-F{x,, y, z)^F{x,, y, z) -F(xo, yo, z) + F(xo, 2/0, z)-F{xo, yo, «o) ; or, applying the law of the mean to each of these differences, and obeerring that F(xo, y©, «o) =0, F(x,y,z)= (x-Xo)F,lxo + e(x-Xo), y, «] + (y - yo) F^ [xo, 2/0 + e\y - yo), «] + (z- Zo) F, [Xo, yo, «o + e"(z - «o)]. Hence F(x, y, «) is of the form ,«v i '•'(*» y» «) = ^ (^' 2/, «) (^ - ^o) W ^ +^(x, y, «)(y-yo) + C'(a;, y, «)(«-«o), where the al)solute values of the functions A(Xy y, «), 5(x, y, ^), C(x, y, «) satisfy the inequalities \A\<Q, \B\<Q, \C\>P for all sets of values of x, y, sj which satisfy (2). Xow let c be a positive number less than I, and -q the smaller of the two numbers / and P€/2Q. Suppose that x and y in the equation (1) are given definite values which satisfy the conditions and that we seek the number of roots of that equation, z being regarded as the unknown, which lie between Zq — e and Zq + «. In ftfl the expression (3), for F(x, y, z) the sum of the first two terms is always less than 2Qyi in absolute value, while the absolute value of the third term is greater than Pt when z is replaced by Zq ± «. From the manner in which tf was chosen it is evident that this last term determines the sign of F. It follows, therefore, that F(x, y, Zq — «) < and F(Xf y, «© -f <) > ; hence the equation (1) has at least one root which lies between «o — « and Zq + e. Moreover this root is unique, •ioce the derivative F, is positive for all values of z between Zq — e and «t 4- «. It is therefore clear that the equation (1) has one and only one root, and that this root approaches Zq as x and y approach x^ and y«, respectively. II, $20] IMPLICIT FUNCTIONS 87 Let us investigate for just what values of the variables x and y the root whose existence we have just proved is defined. Let h be the smaller of the two numbers I and 1H/2U\ the foregoing reason- ing shows that if the values of the variables x and y satisfy the inequalities |x — a"o| < A, |y — yd < ^) ^^^ equation (1) will have one and only one root which lies between z^ — I and Zq -h /. Let 72 be a square of side 2 A, about the point Moix^^ y,,), with its sides parallel to the axes. As long as the point (x, y) lies inside this square, the equation (1) uniquely determines a function of x and y, which remains between Xq^I and Zq-\- I. This function is continuoos, by the above, at the point Mq^ and this is likewise true for any other point Ml of R ; for, by the hypotheses made regarding the func- tion F and its derivatives, the derivative /•\(xi, yi, z^) will be posi- tive at the point M^y since \x-^—Xq\<1, \yi— yn\<lj |«i— «o|<^- The condition of things at ^fl is then exactly the same as at A/^, and hence the root under consideration will be continuous for x^Xi, y = yi. Since the root considered is defined only in the interior of the region /?, we have thus far only an element of an implicit function. In order to define this function out- side of R, we proceed by successive steps, as follows. Let /. be a con- tinuous path starting at the point (^0) //o) and ending at a point (A', }') outside of R. Let us suppose that the variables x and y vary simul- taneously in such a way that the —^ point (x, y) describes the path L. pj^^ ^ If we start at (xq, y^ with the value Zq of z, we have a definite value of this root as long as we remain inside the region R. Let M^ {x^, y{) be a point of the path inside i?, and zi the corresponding value of z. The conditions of the theorem being satisfied for x =Xu y = yi, z = z^, there exists another region Riy about the point M^ inside which the root which reduces to «, for X = Xi, y = yi is uniquely determined. This new region /?, will have, in general, points outside of R. Taking then such a point Aft on the path L, inside R^ but outside R, we may repeat the same con- struction and determine a new region R^, inside of which the solu- tion of the equation (1) is defined; and this process could be repeated indefinitely, as long as we did not find a set of values of X, y, z for whigh F, = 0. We shall content ourselves for the present y) 88 FUNCTIONAL RELATIONS [n,§21 with these statements ; we shall find occasion in later chapters to tieat oertain analogous problems in detail. 8L DeriTttiTes of implicit functions. Let us return to the region Hf and to the solution z = <t>(x, y) of the equation (1), which is a continuous function of the two variables x and y in this region. This function possesses derivatives of the first order. For, keeping y fixed, let us give x an increment Aa;. Then z will have an incre- ment Lx, and we find, by the formula derived in § 20, F{x + Aar, y, « + A«) - F{x, y, z) e AxF,(a: + dAx, y, « -f A«) + A«F,(aj, y, « + ^'A«) = 0. Henoe Ag_ j;(a;+gAa;, y,z + £^z) , Aa? F^{x, y, z + $' Lz) and when Ax approaches zero, ^z does also^ since ;;; is a continuous function of x. The right-hand side therefore approaches a limit, and z has a derivative with respect to xi dx F, In a similar manner we find dz__ F^ dy- fJ Note. If the equation F = is of degree m in «, it defines m functions of the variables x and y, and the partial derivatives dz/dxy dz/dy also have m values for each set of values of the variables X and y. The preceding formulae give these derivatives without ambiguity, if the variable z in the second member be replaced by the value of that function whose derivative is sought. For example, the equation «* -f- y* + «* - 1 = the two continuous functions + VI - aj« - y* and - Vl - «« - y» for Talues of x and y which satisfy the inequality x* -^ y^ < 1. The first partial derivatives of the first are — » — y Vl - ar« - y«* Vl - x« - y«' n,f22] IMPLICIT FUNCTIONS 39 and the partial derivatives of the second are found by merely chang- ing the signs. The same results would be obtained by using the formula dz ^ X cz _ y dx z dy z replacing t by its two values, successively. 22. Applications to surfaces. If we interpret x, y,zaA the Cartesian coordinates of a point in space, any equation of the form (4) F(x,y,«) = represents a surface S. Let (xq, y^, s;^) be the coordinates of a point A of this surface. If the function F, together with its first deriva- tives, is continuous in the neighborhood of the set of values Xq, y^f «„» and if all three of these derivatives do not vanish simultaneously at the point .1, the surface .s' has a tangent plane at A. Suppose, for instance, that F, is not zero for ar = Xq, y = y^j « = «©• Accord- ing to the general theorem we may think of the equation solved for z near the point A , and we may write the equation of the surface in the form z = <i>(x, y\ where <f> (x, y) is a continuous function ; and the equation of the tangent plane at A is Replacing dz/dx and dz /dy by the values found above, the equation of the tangent plane becomes <« (S).<--'-(l).<'-->-(tl<-->=»- If F, = 0, but Fj ^ 0, at A^ we would consider y and z as inde- pendent variables and x as a function of them. We would then find the same equation (5) for the tangent plane, which is also evi- dent a priori from the symmetry of the left-hand side. Likewise the tangent to a plane curve F(x, y) = 0, at a point (x^, y^), is <-- •)(£)/<-'->®).-»' If the three first derivatives vanish simultaneously at the point A, 40 FUNCTIONAL RELATIONS [11,523 the preceding reasoning is no longer applicable. We shall see later (Chapter III) that the tangents to the various curves which lie on the surface and which pass through A form, in general, a cone and not a plane. In the demonstration of the general theorem on implicit functions we assumed that the derivative F^ did not vanish. Our geometrical intuition explains the necessity of this condition in general. For, if jc = but F^ ^ 0, the tangent plane is parallel to the z axis, and a line parallel to the z axis and near the line x = x^^, y =^ y^ meets the surface, in general, in two points near the point of tangency. Hence, in general, the equation (4) would have two roots which both approach Zq when x and y approach x^^ and 2/g, respectively. If the sphere x* + y* + «' — 1 = 0, f or instance, be cut by the line y = 0, X = 1 -f c, we find two values of «, which both approach zero with < ; they are real if c is negative, and imaginary if c is positive. 23. Successive derivatives. In the formulae for the first derivatives, dx" fJ dy fJ we may consider the second members as composite functions, z being an auxiliary function. We might then calculate the successive deriv- atives, one after another, by the rules for composite functions. The existence of these partial derivatives depends, of course, upon the existence of the successive partial derivatives of F(x, y, z). The following proposition leads to a simpler method of determin- ing these derivatives. If several functiona of an independent variable satisfy a relation F = 0, their derivatives satisfy the equation obtained by equating to tiero the derivative of the left-hand side formed by the rule for differ- mHaiimg composite functions. For it is clear that if F vanishes identioally when the variables which occur are replaced by func- tions of the independent variable, then the derivative will also van- ish identically. The same theorem holds even when the functions which satisfy the relation /•' = depend upon several independent variables. Now suppose that we wished to calculate the successive derivatives of an implicit function y of a single independent variable x defined by the relation U.pj IMPLICIT FUNCTIONS 41 We find successively d^F d*F d*F dF S'F , . e^F , . „ S*F „ , ^ 8*F „ e*F . + ^I^y'^" + ¥y^"' = ''' from which we could calculate successively y', y", y'", • ••. Example. Given a function y =/(x), we may, Inversely, consider y as the independent variable and z as an implicit function of j/ defined by the equation y=/{x). If tlie derivative /'(x) does not vanish for the value asot where yo = /(Zo), there exists, by the general theorem proved above, one and only one function of y which satisfies the relation y =/(x) and which takes on the value Xo for y = yo- This function is called the inverse of the function /(z). To cal- culate the successive derivatives z„, z,^, z^, • • • of this function, we need merely differentiate, regarding y as the independent variable, and we get 1 = /'(Z) Zy, = /"(z) (z,)'» + /'(x)z^, = r\x) (z,)« + 3/"(z) x,x^ + /'(z) z/, whence It should be noticed that these formulae are not altered if we exchange z, and /'(z), Xj/i and /"(z), Zy« and /'"(z), • • • , for it is evident that the relation between the two functions y = /(z) and z = ^ (y) is a reciprocal one. As an application of these formulae, let us determine all thoee function* y=.f(x) which satisfy the equation y'y"' - 3y"'' = 0. Taking y as the independent variable and z as the function, Uiis equation becomes But the only functions whose third derivatives are zero are polynomials of at most the second degree. Hence z must be of the form z= Ciy«+ Cty-\- C, where Ci, Cj, Cg are three arbitrary constants. Solving this equation for y, we see that the only functions y = /(z) which satisfy the given equation an of the form ^____„ y = o i Vto + c, 42 FUNCTIONAL RELATIONS [II, §24 when a, ft, c are three arbitrary constants. This equation represents a parabola wboM axis is parallel to the z axis. M. PutUl derivatiyes. Let us now consider an implicit function of two variables, defined by the equation (6) JF'(x,y,«) = 0. The partial derivatives of the first order are given, as we have seen, by the equations To determine the partial derivatives of the second order we need only differentiate the two equations (7) again with respect to x and with respect to y. This gives, however, only three new equations, for the derivative of the first of the equations (7) with respect to y is identical with the derivative of the second with respect to x. The new equations are the following: d^F d^F dz d^F /dzV dF d'^z _ / dx^^ dx dz dy dz^ \dx) "^ dz dx". ~ "' a«F^. a«F dz . d'^FVz . d^Fdzd^z . dF d^z (8) dx dy dx dz dy dy dzjdx dz^ dx dy dz dx dy ' df '^ dydzdy^ dz^ \dy) ^ dz df ~ "' The third and higher derivatives may be found in a similar manner. By the use of total differentials we can find all the partial deriva- tives of a given order at the same time. This depends upon the following theorem : If several functwns Uy Vj w, -" of any number of independent vari- ables x,y,x,"- satisfy a relation F = 0, the total differentials satisfy the relation dF = 0, which is obtained by forming the total differential of Fas if all the variables which occur in F were independent variables. In order to prove this let F(u, v, t^;) = be the given relation between the three functions m, r, w of the independent variables x, y, ^, t. The first partial derivatives of m, v, w satisfy the four equations dudr. dv dx'^ dw dx"^ * «i« ^y ■*■ av ay ■*" dw a^ "" "' maij IMPLICIT FUNCTIONS 43 du ex dv dx dw dx ' du dt dv dt dw dt Multiplying these equations by dxy dy, dXf dt, respectively, and adding, we tind dF dF dF ^du-\--fdv\--fdw = dF=0. du ou ow This shows again the advantage of the differential notation, for the preceding equation is independent of the choice and of the number of independent variables. To find a relation between the second total differentials, we need merely apply the general theorem to the equation rfF=0, considered as an equation between w, v, w^ du, dv, dw, and so forth. The differentials of higher order than the first of those variables which are chosen for independent variables must, of course, be replaced by zeros. Let us apply this theorem to calculate the successive total differ- entials of the implicit function defined by the equation (6), where X and y are regarded as the independent variables. We find cF dF dF (dF , , dF , , dF ,Y^ , dF ^ . and the first two of these equations may be used instead of the five equations (7) and (8) ; from the expression for dz we may find the two first derivatives, from that for d^z the three of the second order, etc. Consider for example, the equation which gives, after two differentiations, Axdx -{-A'ydt/ + A"zdx = 0, Adx*-^ A'di/^-{- A"dz^-h A"zd'z = 0, whence Axdx-^A'ydy, <** = Ir. and, introducing this value of dx in the second equation, we find ^ A {A x^ -\- A "z') dx' -h2AA 'xydx dy 4- A '(A 'y*-\-A "z') d/ 41 FUNCTIONAL RELATIONS [11, §24 Using Mongers notation, we have then Ax A^y A(A7^±A^ _ AA'xy A 'jA'y' -h A"z^) ''■ A"^z* *~ A"^z*' A"^z^ This method is evidently general, whatever be the number of the independent variables or the order of the partial derivatives which it is desired to calculate. JBEampte. Let t =/(x, y) be a function of x and y. Let us try to calculate Um diflerentiaLB of the first and second orders dx and d^x, regarding y and z as the independent variables, and x as an implicit function of tiiem. First of all, we have dz = ^-idx+?^dy. dz dy y and z are now the independent variables, we must set d^y = d^z = 0, and consequently a second differentiation gives = ^dx^ + 2f^dxdy + ^dy^ + ^d:ix. ax* dxdy dy^ dx In Mongers notation, using p, g, r, s, t for the derivatives of f{x, y), these •qoaUonB may be written in the form dz = pdx + qdy^ = rdx^ + 28dxdy + tdy^ +pd^z. Pfom the first we find ^^ dz-qdy ^ P and, fttbttituting this value of dx in the second equation, ^j^^__rd^-^2{pa^qr)dydz-\-(qir-2pq8-{-pH)dy^ The flrat and second partial derivatives of x, regarded as a function of y and S, tharalbra, have the following values : 5? -1 ax _ q dz ""p' dy~ p' — « - I., -^ = qr-p a dix _ 2pq8-pH-q^r «■• p«' dydx p^ ' dy^~ pi As an appllcaUon of theM formula, let us find all those functions /(x. y\ which aaUafy the equation ^ v » i'/ U, In the «|Qaaon i =/(x, y), x be considered as a function of the two inde- varUblMi y and i, the given equation reduces to Xy> = 0. This means n.j2a] IMl'LICIT FUNCTIONS 45 that Xy \b independent of y ; and hence x, = 0(2)« where 0(c) Is an arbitrary function of z. Thia, in turn, may Ik« \vri»t.... in the form ^ [x - y (p[Z)j - 0, which shows that z — y 0(2) is Independent of y. Hence we may write « = y0U) + ^(«), where \J/ (z) is another arbitrary function of 2. It is clear, therefore, that all the functions z =/(x, y) which satisfy the given equation, except those for which /» vanishes, are found by Kolving this last equation for 2. This equation represents a surface generated by a straight line whicli is always parallel to the xy plane. 25. The general theorem. Z^et us consider a system ^f n equations ^i(«i, arj, •••, a-p; t/,, Wj, •■•, wj = 0, (E) , j-p , 1*1, i*„ Xpy Wl, W,, P* Suppose between the n -\- p variables Wi, i^j, ••, w„; ari, x„ • ••, ar^ ^Aa^ ^Aese equations are satisfied for the values a*j = a^, Wj = mJ, •• •, u^ = ?^J; ^Aa^ the functions F, are continuous and possess first partial derivatives which are continuous, in the neighborhood of this system of values; and, finally y that the determinant dui dui du^ A = dui du^ cu^ does not vanish for dui du^ du. Xi = X?, u, = < {i = 1, 2, ...,/>;/: = 1, 2, ..., n). Under these conditions there exists one and only one system of CAm- tinunus functions u^ = <^i(a-i, a*,, • • •, x^), • • , w^ = 4>^(xi, a-,, • •-, x^) which satisfy the equations (E) and which reduce to «?, w5, •••, t/^, /o'-ar, = «•», '•',Xp = xl* •In his paper quoted above (ftn., p. 35) Goursat proves that the same conclosion may be rraohe<l without makin>j any hypotheses whatever repanling the derivatives cFi/txj of th«> functions F, with n>pird to the r's. Otherwise the hyiwtheses remain exactly as state<l above. It is to be notioenl that the later thei»rems r^Farding the existence of the derivatives of tlie (unctions 4> would not follow, however, witboat some assumptions regarding dFi/dxj. The. proof given is ba8e<l on the followiiig 46 FUNCTIONAL RELATIONS [II, §26 The determinant A is called the Jacobian^* or the Functional Deter- minant, of the n functions Fi, F„ • •, F, with respect to the n yari- ables Ui,Ut," •, u,. It is represented by the notation We will begin by proving the theorem in the special case of a system of two equations in three independent variables a;, y, z and two unknowns u and v. (9) Fi(x, y, «, w, v) = 0, (10) Fa(x,y,z,u,v) = 0. These equations are satisfied, by hypothesis, for a; = a;©? y = yo> « = ^o> u = Oq, r = r^, ; and the determinant gFi aF, dFi gFa ^M ^v a^ du does not vanish for this set of values. It follows that at least one of the derivatives dFi/dv, dF^/dv does not vanish for these same values. Suppose, for definiteness, that dFi/dv does not vanish. According to the theorem proved above for a single equation, the relation (9) defines a function v of the variables x, y, «, u, which reduces to Vq for x = Xq, y = y^, z = «q, w = i<^. Replacing v in the equation (10) by this function, we obtain an equation between z, y, Xf and m, ♦(x, y, «, w) = Fa[a;, y, z, w, /(a;, y, z, w)] = 0, toUM: Let /iizuZt,",Xp;uuUt,",u^, •••./«(«!, ««,• -.Kp; "i, t*2. ••. O ben ffmetUnu qfthsn-i-p variables a;, and u^, which, together with the n'^ partial deriva- tktm^i/du^,are continuous near Xi = 0, x^ =0, • •, Xp = 0, u^ =0, •., m„ = 0. If ^n functions fi and the n* derivcUives cfi/dui, all vanish for this system of values^ tkmi tk§ n equations admii one and only one system of soliUions of the form «l«^l(«lt«t. •••.»^), ttt = *«(«!. «i,-,a:p), •., Mn = 0,(a;i,a;a, •••,ajp), ••*•* ^» ^. •••• ♦• <w« oonh'nuotM/uncfton^ o/ fAc p variables Xi, «2, •• •, «p wAicA •II ^pprooM lero a« (A« variables all approach zero. The lemma is proved by means of ••olUol functUmit^-) ==/,(x,,x,, ...,x^; wi--^ 4«-^ ..., ui'«-»>) (t = l, 2, ... whan ^tsO. It to •bown that the suite of functions mS'"> thus defined approaches a ItoilUaff fttaetkm Ui, which I) aatlsfles the given equations, and 2) constitutes the only •olatSoo. Tb* pMMfe from the lemma to the theorem consists in an easy transforma- tiott at tbo •qoAtlooa (E) loto a form similar to that of the lemma. —Trans. . * Jaoom, Ostft'f Journal, Vol. XXII. il,§25] IMl'LICIT FUNCTIONS 47 which is satisfied for a; = ajo, y = y^* « = «o> u = tt^. Now and from equation (9), du du dv du du dv du ' whence, replacing df/du by this value in the expression for d^/dtu, we obtain d^ D(u, v) 'du "^ dFi dv It is evident that this derivative does not vanish for the values ar^,, yo> *oj ^^0* Hen ce^ the equation * = is satisfied when u is replaced by a certain continuous function ti = <f> (x, y, «), which is equal to i/q when x = Xq, y = y^y z = Zq\ and, replacing m by <^ (x, y, 2) in /(x, y, «, w), we obtain for v also a certain continuous function. The proposition is then proved for a system of two equations. We can show, as in § 21, that these functions possess partial derivatives of the first order. Keeping y and z constant, let us give X an increment Ax, and let An and Av be the corresponding increments of the functions u and v. The equations (9) and (10) then give us the equations "(S-)-"(S-')-(^- -■■)-» where c, «', «", rj, rj', rj" approach zero with Ax, Aj/, Ar. It follows that Am Ax When Ax approaches zero, Am and ^v also approach zero ; and henoe c, c', e", rj, rj', rj" do 80 at the same time. The ratio Am /Ax therefore approaches a limit; that is, u possesses a derivative with respect to x : 48 FUNCTIONAL RELATIONS [n,§25 dF\dF^_df\dF\ du dxdvdvdx du dv dv du It follows in like manner that the ratio Av/Aa; approaches a finite limit dv/dx, which is given by an analogous formula. Practically, thaw derivatives may be calculated by means of the two equations dx du dx dv dx ' dx du dx dv dx ' and the partial derivatives with respect to y and z may be found in a similar manner. In order to prove the general theorem it will be sufficient to show that if the proposition holds for a system of (n — 1) equations, it will hold also for a system of n equations. Since, by hypothesis, the functional determinant A does not vanish for the initial values of the variables, at least one of the first minors corresponding to the elements of the last row is different from zero for these same values. Suppose, for definiteness, that it is the minor which corresponds to dP^/dii^ which is not zero. This minor is precisely and, since the theorem is assumed to hold for a system of (n — 1) equations, it is clear that we may obtain solutions of the first (n — 1) of the equations (£) in the form Ut = ^i(xi, x,, ...,0;^; w,), ..., w,.i = <^„_x(aJi, Xa, ••, Xp; u^), where the functions <^, are continuous. Then, replacing Ui, •-•, i^„_, by the functions </»,, • • -, <^^_, in the last of equations (E), we obtain a new equation for the determination of w., It only remains for us to show that the derivative d^/du^ does not vanish for the given set of values a-«, xj, • • -, xj, wj ; for, if so, we can toWe this laet equation in the form where ^ is continuous. Then, substituting this value of u^ in 4it ••, ^,.,, we would obtain certain continuous functions foi 11,525] IMi'LiLil FUNCTIONS 49 ^11 ^1 '"> ^n-i ^^* ^^ order to show that the derivative in ques- tion does not vanish, let us consider the equation The derivatives ^<^,/^m„, 3<^,/^m,, ••, ^<^,_,/3m, are given by the (n — 1) equations (12) dui du^ ^u^-i ^w, ^w« ' and we may consider the equations (11) and (12) as n linear equa- tions for d4>i/du^j • ••, b<t>^_i/du^f d^/du^j from which we find d^ D(F,, F„ ■■•, F,.,) ^ Z)(F„ F>, .-, FJ du^ D{ic^, Ws, ..., u^.{) D(ui, Ui, •••, w,) It follows that the derivative d^ /du^ does not vanish for the initial values, and hence the general theorem is proved. The successive derivatives of implicit functions defined by several equations may be calculated in a manner analogous to that used in the ease of a single equation. When there are several independent variables it is advantageous to form the total differentials, from which the partial derivatives of the same order may be found. Consider the case of two functions u and v of the three variables Xy y, z defined by the two equations F(x, y, «, M, v) = Oy *(ar, y, z, ?/, r) = 0. The total differentials of the first order du and dv are given by the two equations ox oy cz cii cv Likewise, the second total differentials d*u and d*v are given by the equations 60 FUNCTIONAL RELATIONS [n,§26 and so forth. In the equations which give d'*u and d^v the deter- minant of the coefficients of those differentials is equal for all values of n to the Jacobian Z>(F, *)/D(?^, v), which, by hypothesis, does not ▼anish. t6. iBTtnioii. Let ui, tit, • • •, u^ be n functions of the n independent vari- *if ««» • • -I ^1 such that the Jacobian D(ui, 1*2, •• » w„)/D(xi, x^, •••,«„) not Tuiish identically. The n equations (IS) \ "* ~ ^^**' **'*■' ^^' ^* ~ 02(«1, «2, • • • » «n), • • • , inversely, asi, Xj, • • • , «„ as functions of wi, U2, • • • , Wn- For, taking any of values x?, xj, • • •, xj, for which the Jacobian does not vanish, and denoting the corresponding values of Ui, v^, • • •, t<M by u^, u^, • • ., wj, there exists, according to the general theorem, a system of functions «t = fl(Ul, U,,.-.,li„), Xi = rp2{UuU2,-",Ur,), .-•, X„ = ^„(Ui, Wj, • • • , Mn), which satisfy (18), and which take on the values xj, xj, • • • , xj, respectively, when ui = Uj, • • • , tt» = mJ. These functions are called the inverses of the func- tions 01, 0s, • ••, 011, and the process of actually determining them is called an inversion. In order to compute the derivatives of these inverse functions we need merely apply the general rule. Thus, in the case of two functions u=/(x, y), « = 0(x, y), if we consider u and v as the independent variables and x and y as inverse funcUona, we have the two equations du •'£"* 'J-dy dv = 50 , , 50 , dz. dfdiP .'Id. c/di>' dy = dx dx dfd<f> dfdi> dx dy dy dx dx dy dy dz Bnal ly, the formulas d0 a/ 9z dy dz _ dv ~ ay 9u a/«0 bzdy a/a0 ' dy dz djdj dz dy dy dz II, §27] IMPLICIT FUNCTIONS 61 -£* a/ dx ^ dy dx dx dy ey dx 27. Tangents to skew curves. Let us consider a curve C repre- senteil bv tlie two e(juatioiis (14) (/'x(a.,y,^)=U, lFt(x, y, «)=0; and let x^^ y^, Zq be the coordinates of a point A/© of this curve, such that at least one of the three Jacobians dF\dF\ _dF\dJ\ dF\ dF\ _ dF\ dF\ dF^ gF, cF^ dF^ dy dz dz dy dz dx dx dz dx dy dy dx does not vanish when x, y, z are replaced by a-Q, yo> ^o> respectively. Suppose, for definiteness, tliat />>(Fi, F^/ D{y, z) is one which does not vanish at the point 3/o. Then the equations (14) may be solved in the form y = 4>(x), z = ^(x), where <f> and ^ are continuous functions of x which reduce to y^ and «o, respectively, when x = Xq. The tangent to the curve C at the point Mq is therefore represented by the two equations X-x, ^ Y-y, ^ Z-z^ 1 ^\x,) f(xo)' where the derivatives <^'(a-) and •/''(•^) ^^y ^® found from the two equations g+if *'<'>-!? ♦■<-)-«. S^ + |^*V).gf(.)... In these two equations let us set a? = x,,, y = y^, « = «o, and replace <A'(Xo) and f(Xo) by ( r - yo) / ( A' - x^) and (Z - ;ro)/(A - Xo), respectively. The equations of the tangent then become (16) 53 FUNCTIONAL RELATIONS [II, §28 L ^(y> *) Jo L ^(«. ^) Jo L ^(*>y) Jo The geometrical interpretation of this result is very easy. The two equations (14) represent, respectively, two surfaces S^ and ^a, of which C is the line of intersection. The equations (15) represent the two tangent planes to these two surfaces at the poirt Mq ; and the tangent to C is the intersection of these two planes. The formulae become illusory when the three Jacobians above all TanUh at the point M^. In this case the two equations (15) reduce to a single equation, and the surfaces 5i and ^2 are tangent at the point 3/o. The intersection of the two surfaces will then consist, in general, as we shall see, of several distinct branches through the point J/«. II. FUNCTIONAL DETERMINANTS 28. Fundamental property. We have just seen what an important r61e functional determinants play in the theory of implicit functions. All the above demonstrations expressly presuppose that a certain Jacobian does not vanish for the assumed set of initial values. Omitting the case in which the Jacobian vanishes only for certain particular values of the variables, we shall proceed to examine the very important case in which the Jacobian vanishes identically. The following theorem is fundamental. Ta^ Ui, f/,, • • • , u^ be n functions of the n independent variables jt|, X,, •••, X,. In order that there exist between these n functions a relation H (ui, w,, • • •, w,) = 0, which does not involve explicitly/ any of the variaJjles Xi, x^t " - f x^, it is necessary and sufficient that the fwMtional determinant, D(ur,Ui,'",u„) D{XijX^y-,X^) should vanish identically. In thj first place this condition is necessary. For, if such a rela- tion n(it„ w„ . . ., mJ = exists between the n functions u^, u^, - • ., w^, the following n equations, deduced by differentiating with respect to each of the x*8 in order, must hold : i,ij.. FUxVCTlONAL DETKKMINANTS 58 dui dz\ du^ dxi cn,^ rr^ * dui dx^ du^ dx^ du^ dx^ "" ' and, since we cannot have, at the same time, — = — = =?II-o since the relation considered would in that case reduce to a trivial identity, it is clear that the determinant of the coefficients, which is precisely the Jacobian of the theorem, must vanish.* The condition is also sufficient. To prove this, we shall make use of certain facts which follow immediately from the general theorems. 1) Let u, V, w be three functions of the three independent variables Xj y, z, such that the functional determinant D(u, v, w)/D(x, y, z) is not zero. Then no relation of the form Xdu -{• fxdv + vdw = can exist between the total differentials du, du, dwy except for X = /x = V = 0. For, equating the coefficients of dx, rfy, dz in the foregoing equation to zero, there result three equations for X, /x, v which have no other solutions than X = /i = v = 0. 2) Let CO, w, V, IV be four functions of the three independent variables ar, y, «, such that the determinant />(»«, r, w)/ D(x, y, z) is not zero. We can then express x, i/, z inversely as functions of u, V, w'^ and substituting these values for ar, y, z in w, we obtain a function O) = * (w, V, w) of the three variables w, v, w. If by any process whatever we can obtain a relation of the fonn (16) dia = Pdu+ Qdv + Rdw *A8 Professor Osgood has pointed out, the reasoning here suppoMS that the partial dcrivative.s dll/tu^ , c*II /tuf^, • • • , ?n /^u« do not all vanish simultaneously for any system of values which cause 11 (ti|, u^, • • -, u„) to vanish. This supposition \h certainly justified when the relation n = is solved for one of the variables u,. 54 FUNCTIONAL RELATIONS [li,§28 between the total differentials d<o, du, dv, dw, taken with respect to the mdependetU variables x, y, «, then the coefficients P, Q, R are equaly reepeetiwelyt to the three first partial derivatives of^{u, v, w) : For, by the rule for the total difFerential of a composite function (5 16), we have dut = T- du -{- -^ dv -\- -^ dw'j du ov ow and there cannot exist ady other relation of the form (16) between du, du, du, dw, for that would lead to a relation of the form \ du -{- fi dv + V dw = 0, where X, /i, v do not all vanish. We have just seen that this is impossible. It is clear that these remarks apply to the general case of any number of independent variables. Let us then consider, for definiteness, a system of four functions of four independent variables X = Fi(x, y, z, t), Y = F,(x, y, z, t), Z z= F^{x, y, z, t), r = F,{x, 7j, z, t), (17) where the Jacobian D(Fi, F,, F^, F^)/D(x, y, z, t) is identically zero by hypothesis ; and let us suppose, first, that one of the first minors, say /)(Fi, F,, F^)/D(x, y, z), is not zero. We may then think of the first three of equations (17) as solved for x, y, z as functioDB of X, Y,Z,t; and, substituting these values for x, y, z in the last of equations (17), we obtain T as a function of X, Y,Z,t: W T^4f(X,Y,Z,t). We proceed to show that this function * does not contain the vari- able t, that is, that d^/dt vanishes identically. For this purpose let ui consider the determiuaut II.§*-«J FUNCTIONAL DETERB4INANTS 55 A = dFy' dF\ dF\ d^ dy dx gF, dp; dFt dx dy dz dF\ dF\ e^ dx dy dz dj\ dj\ d¥\ dx dy dz dX dY dZ dT If, in this determinant, dXy dVj dZ, dT be replaced by their values and if the determinant be developed in terms of rfx, rfy, dz, dty it turas out that the coefficients of these four differentials are each zero ; the first three being determinants with two identical columns, while the last is precisely the functional determinant. Hence A = 0. But if we develop this determinant with respect to the elements of the last column, the coefficient of dT'is not zero, and we obtaia a relation of the form dT=PdX-ir QdY-\- RdZ. By the remark made above, the coefficient of dt in the right-hand side is equal to d<if/dt. But this right-hand side does not contain dtf hence d<P/dt = 0. It follows that the relation (18) is of the form r = <i»(x, r,z), which proves the theorem stated. It can be shown that there exists no other relation, distinct from that just found, between the four functions A', 1', Z, 7', independent of Xj y, z, t For, if one existed, and if we replaced T by *(J^, K, Z) in it, we would obtain a relation between A', Y, Z of the form n(A', K, Z)=0, which is a contradiction of the hypothesis that D{X^ F, Z) / D(x^ y, z) does not vanish. Let us now pass to the case in which all the first minors of the Jacobian vanish identically, but where at least one of the second minors, say Z)(F,, Ft)/D(x, y), is not zero. Then the first two of equations (17) may be solved for x and y as functions of X. y, z, t, and tlie last two become Z = *, (A, }', z, 0, T -. *, (A, r, z, t). 56 FUNCTIONAL RELATIONS [il.§28 On the other hand we can show, as before, that the determinant ^-II p dX dx dy ^Jl ^Jl dY dx dy !^ p dz dx oy vanishes identically ; and, developing it with respect to the elements of the last column, we find a relation of the form dZ^PdX+QdY, whence it follows that »-' dt dt "' Id like manner it can be shown that and there exist in this case two distinct relations between the foui functions X, Y, Z, T, of the form Z = *i(X, F), T=^,(X,Y). There exists, however, no third relation dittinct from these two; for, if there were, we could find a relation between X and F, which would be in contradiction with the hypothesis that 2)(X, Y)/D(xy y) if not zero. Finally, if all the second minors of the Jacobian are zeros, but not all four functions Xy Z, F, T are constants, three of them are functions of the fourth. The above reasoning is evidently general. If the Jacobian of the n functions Fi, F^, •• •, F„ of the n independ- ent variables Xi, Xj, • ••, a;,, together with all its (w — r -f- 1)- rowed minors, vanishes identically, but at least one of the (n — r)- rowed minors is not zero, there exist precisely r distinct relations between the n functions ; and certain r of them can be expressed in terms of the remaining (n — r), between which there exists no relation. The proof of the following proposition, which is similar to the above demonstration, will be left to the reader. The necessary and tMfficitnt condition that n functions of n -\- p independent variables be conmeeted by a rf lot ion which does not involve these variables is that one of the Jaeobians of these n functions^ with respect ^o any n II, J 28] FUNCTIONAL DETERMINANTS 57 of the independent variableSy should vanish identically. In par- ticular, the necessary and sufficient condition that two functions /'\(ar,, Xj, ••, X,) and /'\(xi, x,,--, x,) should be functions of each other is that the corresponding partial derivatives dF^/dxi ^^^ dFt/dXi should be proportional. Note. The functions F,, Fj, • • •, F. in the foregoing theorems may involve certain other variables yi, y,, •••, y^, besides Xj, x,, ••-, x,. If the Jacobian 7^(Fi, F,, ••, F^)//>(xi, x,, •••, x^) is zero, the functions F,, Fj, •••, F„ are connected by one or more relations which do not involve explicitly the variables Xj, x,, •••, x„, but which may involve the other variables yi,.ys, •••, y,„. Applicationa. The preceding theorem is of great importance. The funda- mental property of the logarithm, for instance, can be demonstrated by means of it, without using the arithmetic definition of the logarithm. For it is proved at the beginning of the Integral Calculus that there exists a function which Is defined for all positive values of the variable, which is zero when x = 1, and whose derivative is \/x. Let/(x) be this function, and let "=/(Jj)+/(y). o = xy. Then X y V £ 0. Henc& there exists a relation of the foriu /(aj)+/(y) = 0(«y); and to determine <t> we need only set y = 1, which gives f{x) = ^ (z). Hence, since x is arbitrary, f{x)-^f{y)=f{xy). It is clear that the preceding definition might have led to the discovery of the fundamental properties of the logarithm had they not been known before the Integral Calculus. As another application let us consider a system of n t-ciuations in n unknowns (19) Fi(ui, ua, •••, u„) = J^i, Ft{uu U2, ••., u,) = //,, I where J/i, Hs, •••, Hn ft^e constants or functions of certain other variables 3:1, Xj, •.., Xm, which may also occiu: in the functions F,. If the Jacobian ^(Fi, Fj, • . ., F„)/D(ui, Ua, • •, u^) vanishes identically, there exist between the n fimctions F, a certain number, say n — ilr, of distinct relations of the form Ft+i = ni(Fi, • . ., Ft), . . . , F, = n„_*(Fi, . , Ft). 68 FUNCTIONAL RELATIONS [II, § 29 In order that the eqaations (10) be compatible, it is evidently necessary that Ht^i = ni(£ri, . . ., jJt), . . ., JET, = nn-t(Hi, • • • , ff*), and, if this be true, the n equations (10) reduce to k distinct equations. We have then the same cases as in the discussion of a system of linear equations. 29. Another property of the Jacobian. The Jacobiau of a system of n functions of n variables possesses properties analogous to those of the derivative of a function of a single variable. Thus the preceding theorem may be regarded as a generalization of the theorem of § 8. The formula for the derivative of a function of a function may be extended to Jacobians. Let Fj, Fj, • ••, F„ be a system of n func- tions of the variables Wi, Wj, •••, u„, and let us suppose that Wj, U2, .••, tt, themselves are functions of the n independent variables a^i, jBg, • • ., X,. Then the formula D(F,, F„ ■■, f\) __ D(Fiy Fa, >.., F,) D(uu u^,"-, u,) /)(xi, x,, ..., iTj D{u^,u^y"-,u^) D(xi, Xa, ...,x„) follows at once from the rule for the multiplication of determinants and the formula for the derivative of a composite function. For, let us write down the two functional determinants dui dFi dx. dx. dF^ . J dui dx„ where the rows and the columns in the second have been inter- changed. The first element of the product is equal to dFi dui dFi dut dui dxi du^ dxi du^ dxi that is, to dFi/dxiy and similarly for the other elements. •0. BMdaas. Let /(x, y, 2) be a function of the three variables x, y, «. Then the fuiicUooal det«nninant of the three first parUal derivatives df/dx, df/dy, V/^. h = 5/ j^f _av 3x* dxdy dxdz dxdy V dydz a*/ y/ y/ dxdt dydz dz* 11. §30] FUNCTIONAL DETERMINANTS 69 is called the Hessian of /(z, y, z). The Heasian of a function of n rariablet Is defined in like manner, and plays a rdle analogous to that of the second deriTa- tive of a function of a single variable. We proceed to prove a remarkable invariant property of this dei«ruiiuaut. Let us suppose the independent Tari> ables traiuiformed by the linear subetitution where X, F, Z are the transformed variables, and or, /9, 7, • • • , y^ are constants such that the determinant of the substitution. (1»0 a ^ 7 a" ^ 7' a" r 7' is not zero. This substitution carries the function /(x, y, z) over into a new function F(A', F, Z) of the three variables X, F, Z. Let // (X, 1', Z) be the Hessian of this new function. We shall show that we have identically H{X, r, Z) = A«A(x.y,z), where 2, y, z are supposed replaced in A(z, y, z) by their expressions from {W). For we have /aF SF aF\ T)/££ cF iF\ ^_ \dx' dY' izj _ \dx' ar' ez/ d(2, y, z) . D(X, r, Z) ~ D(x, y, 2) I>(X, r, Z) ' and if we consider tf/dx^ ^/Z^, ^//^«f for a moment, as auxiliary variables, we may write eF £F eF^ ax' ar' eZ/ ~ Vex_£y__£2/ D(x, y, z) D(x,y, z) 'D(X, F, Z) £r = \ax' iy' dz) ygx* ay* c^^/ \ax ay az / But from the relation F(X, F, Z) =/(x, y, 2), we find 3/ ZF a/ . ,a/ . , -— = a— ■ha'— -\-a' dX dx dy dz BY ^ dx ^ dy ^ dz whence dZ ^ bx dy tf < dF ax' dF dF dZ and hence, finally, \dx dy czj P U = Ah 7 D(x, y, 2) D(x, r, z) It is clear that this theorem is general. dz a" a" = A«*. A; dO FUNCTIONAL RELATIONS [ll, §30 Let vm now consider an application of this property of the Hessian. Let /(z, y) = ou^ + 3ftx«y + 3cxy2 + dy* bt a glrwi Wnmry cubic form whose coefficients a, 6, c, d are any constants. Hmh, neglecting a numerical factor, I « + 6y *« + ^ I = (ac - 62) x2 + (od - 6c) xy + (M - c2) y2, l&t + cycx + di/l and the Heeelan is seen to be a binary quadratic form. First, discarding the OMe in which the Hessian is a perfect square, we may write it as the product of two Unew factors : h = (mx + ny) (pz + qy). If, DOW, we perform the linear substitution mx-^ny = X, px + qy = Yj the fonn/(x, y) goes over into a new form, F(X, F) = AX» + 3BX2r+ 3 CXY^ + DF«, wbaee Heeeian is H{X, T) = {AC-&)X^ + {AD - BC) XY + {BD - C^) Y% and this must reduce, by the invariant property proved above, to a product of the form KXY. Hence the coefficients A, B, C, D must satisfy the relations Bi-AC = 0, BD-C^ = 0. If one of the two coefficients £, C be different from zero, the other must be so, and we shall have C B BC BC whence F(X, F), and hence /(x, y)\ will be a perfect cube. Discarding this parUcnlar case, it is evident that we shall have B= C = 0; and the polynomial F(X, F) will be of the canonical form AX* + BY*. Hanoe the reduction of the form /(z, y) to its canonical form only involves the efrintloB of an equation of the second degree, obtained by equating the Hessian of the giren form to zero. The canonical variables X, Y are precisely the two faetofi of the Heeelan. It is eesy to see, in like manner, that the form /(x, y) is reducible to the form AX* -i- BX* F when the Hessian is a perfect square. When the Hessian van- iehai identically /(z, y) is a perfect cube : /(x, y) = (crx + /Jy)«. II, J 31] TRANSFORMATIONS 61 III. TRANSFORMATIONS It often happens^ in many problems which arise in Mathematical Analysis, that we are led to change the independent variables. It tlierefore becomes necessary to be able to express the derivatives with respect to the old variables in terms of the derivatives with respect to the new variables. We have already considered a problem of this kind in the case of inversion. I^et us now consider the question from a general point of view, and treat those problems which occur most frequently. 31. Problem I. Let y be a function of the independent variable jc, and let t be a new independent variable connected with x by the relation X = <f>(t). It is required to express the successive derivatives ofy with respect to x in terms of t and the successive derivatives of y with respect to t. Let y=f(x) be the given function, and F(t) =/[<^(0] the func- tion obtained by replacing x by <f>(t) in the given function. By the rule for the derivative of a function of a function, we find di/ du ,, ^ whence dy — ^^ — ^''* ^'~ 4>\t)~ 4>\t) This result may be stated as follows : To find the derivative of y with respect to x, take the derivative of that function with respect to t and divide it by the derivative of x with respect to t. The second derivative d^y /dx^ may be found by applying this rule to the expression just found for the first derivative. We find : d d^y ^ dt^^'^ _ y„<t>'(t)-y,4>"(t) , dx^ <f>xt) [<^'(or ' and another application of the same rule gives the third derivative 52 FUNCTIONAL RELATIONS LU,§32 or, perfonning the operations indicated, The remaining derivatives may be calculated in succession by ropeated applications of the same rule. In general, the nth deriva- tive of y with respect to x may be expressed in terms of <^'(^), <f>"(t)j •••> ♦^■^(O' *^^ *^® ^^^^ ^ successive derivatives of y with respect to t, ' These formulae may be arranged in more symmetrical form. Denoting the successive differentials of x and y with respect to ^ by dx, dy, d*x, d^y, •••, d'^x, dry, and the successive derivatives of y with respect to x by y', y", •••, y^''^ we may write the preceding formulae in the form (20) „ dxd^y — dyd^x y " dx^ ,„ d*ydx^-Sd^ydxd^x-hS dy (d^xY - - dyd^xdx ^ - dx^ The independent variable t, with respect to which the differentials on the right-hand sides of these formulae are formed, is entirely arbitrary ; and we pass from one derivative to the next by the recurrent formula ^ dx the second member being regarded as the quotient of two differen- tials. 32. ApplicationB. These formulae are used in the study of plane curves, when the coordinates of a point of the curve are expressed in terms of an auxiliary variable t. "^ aj=/(0> y = <f>(t)- In order to study this curve in the neighborhood of one of its points it is necessary to calculate the successive derivatives y', y", -of y with respect to « at the given point. But the preceding formulae give us precisely these derivatives, expressed in terms of the succes- Stire derivatives of the functions f{t) and <^ (t), without the necessity II, § 32] TRAN8FOR^f ATTON8 Qg of having recourse to the explicit • a of y as a function of «, which it might be very ditticult, i> y, to obtain. Thus the first formula gives the slope of the tangent. The value of y" occurs in an impor- tant geometrical concept, the radius of eurvature, which is given by the formula |y"l which we shall derive later. In order to find the value of /?, when tlie coordinates x and i/ are given as functions of a parameter t, we need only replace i/' and y" by the preceding expressions, and we find \dxd'^y —dyd^x\ where the second member contains only the first and second deriva- tives of X and y with respect to t. The following interesting remark is taken from M. Bertrand^a Traiii de Calcul diff^rentiel et integral (Vol. I, p. 170). Suppose that, in calculating some geometrical concept allied to a given plane curve whose coordinates x and y are supposed given in terms of a parameter t, we had obtained the expression F(x, y, dx, dy, dPx, dl^y, • • , d-x, d-y), where all the differentials are taken with respect to t Since, by hypothesis, this concept has a geometrical significance, its value cannot depend upon the choice of the independent variable t. But, if we take x = ^ we shall have dx = d£, cPx = dfix = • • = d«x = 0, and the preceding expression becomes which is the same as the expression we would have obtained by supposing at the start that the equation of the given curve was solved with respect to y in the form y = <J>(x). To return from this particular case to the case where the inde- I^endent variable is arbitrary, we need only replace y', y", • • by their values from the formulas (20). Performing this substitution in we should get back to the expression F(x, y, dx, dy, cPx, d«y, • • •) with which we started. If we do not, we can assert that the result obtained is incorrect. For example, the expression dxcPy -f dyd^x (dx« -f dy")' 64 FUNCTIONAL RELATIONS [ii,§33 CAiuot have any geometrical significance for a plane curve which is independent of the choice of the independent variable. For, if we set x = t^ this expression redooM to ^"/(l + y'*)* ; and, replacing y" and y" by their values from (20), we do not get back to the preceding expression. 33. The formulffl (20) are also used frequently in the study of differential equations. Suppose, for example, that we wished to determine all the functions y of the independent variable x, which satisfy the equation (21) (i_.^g_.J + „., = o, where n is a constant. Let us introduce a new independent variable t, where x = cos t. Then we have dy _ dt ~r~ — > ax — sin t d}y dx^ dt"^ "dt. and the equation (21) becomes. after the substitution, (22) ePv J, + »v - 0. It is easy to find all the functions of t which satisfy this equation, for it may be written, after multiplication by 2 dy/dt, whence ^SS--/i=l[(S)v»v]=o, where a U an arbitrary constant. Consequently g = „V^T37, -r- - - » = 0. U. § -M] TRANSFORMATIONS 66 The left-hand side U the derivative of arc sin (y /a) — nt. It follows that this difference must be another arbitrary constant b, whence y = a8in(n^ 4- b), which may also be written in the form 1/ = A Binnt -^ B cos nt Returning to the original variable x, we see that all the functions of X which satisfy the given equation (21) are given by the formula y = A sin (n arc cos x)-^ B cos (n arc cos x), where A and B are two arbitrary constants. 34. Problem II. To evert/ relation between x and y there corresponds, by means of the transformation x =f{tj u), y =: ifi(^tf w), a relation between t and u. It is required to express the derivatives of y with respect to x in terms of tj u, and the derivatives of u with respect to t. This problem is seen to depend upon the preceding when it is noticed that the formulae of transformation, give us the expressions for the original variables x and y as func- tions of the variable t^ if we imagine that u has been replaced in these formulae by its value as a function of ^. We need merely apply the general method, therefore, always regarding x and y as composite functions of ty and w as an auxiliary function of t. We find then, first, dy dy dx d<i> d<i> dt "^ du du dt dx = ; dt ' dt cf.df ct "^ du 7Z' dt and then d^y __ d (dy\ dx 'dx^~"dt \dx/ ' 'di ' or, performing the operations indicated, \H hi dtl\_H^ hi ct dt du\dt ) eudt^J \rt cu dl l[_ £t^ J \ft "*" cH di) ^ FUNCTIONAL RELATIONS [II, §35 In ^neral, the nth derivative y<"> is expressible in terms of t, u, and the derivatives du/dt, d^u/d^, • -, d^u/dt\ Suppose, for instance, that the equation of a curve be given in polar codrdinates p =/(o>). The formulae for the rectangular coor- dinates of a point are then the following : X = p cos 0), y = p sin <i). Let p', p", . • • be the successive derivatives of p with respect to w, considered as the independent variable. From the preceding formulae we find dx = cos Hi dp — p sin o) c^w, dtj = 8ija.n)dp + p cos w e?o», d^x = coscD d^p — 2 sino) dm dp — p cos to c?o)^, d^y =^ sin a> cPp + 2 cos (o dot dp — p sin w c?<u^, whence dx^ + d}/ = dp^-hp^dio^, dx d^y — dij d^x = 2 d(o dp^ — p dto d^ p -h P^ d<oK The expression found above for the radius of curvature becomes R = ± p^4-2p'^-pp' 35. Transformations of plane curves. Let us suppose that to every point m of a plane we make another point M of the same plane cor- respond by some known construction. If we denote the coordinates of the point m by (x, y) and those of M by (X, F), there will exist, in general, two relations between these coordinates of the form (23) X=f{x,y), Y=<t>(x,y). These formulae define a point transformation of which numerous examples arise in Geometry, such as projective transformations, the transformation of reciprocal radii, etc. When the point m describes a curve c, the corresponding point M describes another curve C, whose properties may be deduced from those of the curve c and from the nature of the transformation employed. Let y\ y", -be the suc- oeisive derivatives of y with respect to x, and F', F", • • • the succes- •ive derivatives of Y with respect to X. To study the curve C it is necessary to be able to express F', F", -in terms of x, y, y\ y"t"'» This is pre<u8ely the problem which we have just discussed ; and we find 11. §36] TRANSFORMATIONS 67 dY dx dX dx dY' dx_ dX dx dx^ dy^ and so forth. It is seen that F' depends only on «, y, y\ Hence^ if the transformation (23) be applied to two curves r, c\ which are tangent at the point (a^, y), the transformed curves C, C will also be tangent at the corresponding point {X^ Y). This remark enables us to replace the curve c by any other curve which is tangent to it in questions which involve only the tangent to the transformed curve C, Let us consider, for example, the transformation defined by the formula3 h'^x __ hhj X = :' + y' Y = -fy' which is the transformation of reciprocal radii, or inversion^ with the origin as pole. Let vi be a point of a curve c and M the cor- responding point of the curve C. In order to find the tangent to this curve C we need only apply the result of ordinary Geometry, that an inversion carries a straight line into a circle through the pole. Let us replace the curve c by its tangent mt. The inverse of mt is a circle through the two points M and O, whose center lies on the perpendicular Ot let fall from the origin upon vit. The tangent MT to this circle is perpendicular to AM, and the angles Mmt and mMT are equal, since each is the complement of the angle viOL The tangents mt and MT are therefore antiparallel with respect to the radius vector. 36. Contact transformations. The preceding transformations are not the most general transformations which carry two tangent curves into two other tangent curves. Let us suppose that a point M is determined from each point m of a curve c by a construction Fio. 5 68 FUNCTIONAL RELATIONS [II, §36 which depends not only upon the point m, but also upon the tangent to the curve c at this point. The formulaB which define the trans- formation are then of the form (24) X =/(x, y, y'), Y = 4> (x, y, y') ; and the slope Y' of the tangent to the transformed curve is given by the formula dY _ dx^ dy^^dy'^ dx^ dy^ ^ dy'^ In general, F' depends on the four variables x, y, y\ y"; and if we apply the transformation (24) to two curves c, c' which are tangent at a point (x, y), the transformed curves C, C will have a point (JT, F) in common, but they will not be tangent, in general, unless y" happens to have the same value for each of the curves c and c'. In order that the two curves C and C should always be tangent, it is necessary and sufficient that F' should not depend on y"; that is, that the two functions /(x, y, y') and <^ (x, y, y ') should satisfy the condition a//a^ a^ \ a^/V a/ A. dy\dx^ dy^j dy\dx^dyy ) In case this condition is satisfied, the transformation is called a contact transformation. It is clear that a point transformation is a particular case of a contact transformation.* Let us consider, for example, Legendre's transformation, in which the point 3^, which corresponds to a point (x, y) of a curve c, is given by the equations a: = y', F = xy' - y ; from which we find ■> dX y" ~ ■ .«., which shows that the transformation is a contact transformation. In like manner we find F" dY' dX dx y"dx 1 ylii dY" dX y-' *J*T**'T *"** Ampfcw gave many examples of contact transformations. Sophus see in particular his Geornet Vorledungen iiber Dynamik. U«4«Taloped the general theory in various works; see in particular his Geojuetrie d$r Btrahrunattran^onnatioMn. See also Jacobi II. $37] TRANSFORMATIONS 69 and 80 fortli. From the preceding formulae it follows that which shows that the transformation is involutory.* All these prop- erties are explained by the remark that the point whose coordinates are -Y = y', Y = xi/' — y is the pole of the tangent to the curve c at tlie point (x, y) with respect to the parabola x* — 2 y = 0. But, in general, if 3/ denote the pole of the tangent at w to a curve c with respect to a directing conic 2, then the locus of the point M is a curve C whose tangent at M is precisely the polar of the point m with respect to 2. The relation between the two curves c and C is therefore a reciprocal one ; and, further, if we replace the curve c by another curve c', tangent to c at the point m, the reciprocal curve C will l)e tangent to the curve C at the point M. Pedal curve.s. If, from a fixed point in the plane of a curve c, a perpen- dicular OM be let fall upon the tangent to tlie curve at the point m, the locus of the foot 3/ of this perpendicular is a curve C, which is called the pedal of the given curve. It would be easy to obtain, by a direct calculation, the coordinates of the point M, and to show that the trans- formation thus defined is a contact transfor- mation, but it is simpler to proceed as follows. I^et us consider a circle y of radius H, de- scribed about the point O as center; andlet»ni be a point on 03/ such that Omi x 0M= RK The point mi is the pole of the tangent mt with respect to the circle; and hence the transformation which carries c into C is the result of a transformation of reciprocal po- lars, followed. by an inversion. "When the point /n describes the curve c, the point mi, the pole of mt, describes a curve Ci tangent Fio. 6 to the polar of the point m with respect to the circle 7, that is, tangent to the straight line mi<i, a perpendicular let fall from mi upon Ojh. The tangent 3f Tto the curve C and the tangent mi^i to the curve Ci make equal angles with the radius vector OniiM. Hence, if we draw the normal MA, the angles AMO and A OM are equal, since they are the comple- ments of equal angles, and the point A is the middle point of the line Om. It follows that the normal to the pedal is found by joining the point 3f to the center of the line Om. 37. Projective transformations . Every function y which satisfies the equation V" = is a linear function of x, and conversely. But, if we subject x and y to the projective transformation *That is, two suocessive applications of the transformation lead us back to the original coordinates. — Trans. 70 FUNCTIONAL RELATIONS [II, $38 aX + 6r+c a' X -\- h' Y ■\- cT z = a^'X-k-lf'Y-^cf' a"X-\-h"Y + &' % straight line goes over into a straight line. Hence the equation y" = should d^T/dX* = 0. In order to verify this we will first remark that the projective transformation may be resolved into a sequence of particular transformations of simple form. If the two coefficients a^' and y are not both lero, we will set Xi = a"X -\- h" Y + c" \ and since we cannot have at the same time aft" - ha" = and a'h" — h'a.*' = 0, we will also set Fi = a'X + 6' F + c', on the supposition that a'h" — h' a" is not zero. The preceding formulas may then be written, replacing X and Y by their values in terms of Xx and Fi, in the form ^1 Xx ^ Xx Xx It follows that the general projective transformation can be reduced to a succession of integral transformations of the form x = aX+6F+c, y = a'X + 6'F + c', combined with the particular transformation 1 F this latter transformation , we find dx XY'- F -1 * X2 ~ F - XY\ and y" = ^ = - XY'\- X^\ = X8 F''. dx ' Likewise, performmg an integral projective transformation, we have dx a + &F' y// _ ^' _ {ab'-ba')Y'^ dx {a-\-bYy In each case the equation y" = goes over into F" = 0. We shall now consider functions of several independent variables, and, for dtflniteoess, we shall give the argument for a function of two variables. 38. Problem in. Let a> =/(x, y) be a function of the two independ- ent variables x and y, and let u and v be two new variables connected mth the old ones by the relations ap = <^(tt, v), y = \ff(u, v). h is required to express the partial deHvatiues of <u tvith respect to the variables xandy in terms of w, v, and the partial derivatives of <u with '"* to u and v. n, J 38] TRANSFORMATIONS 71 Let a» = F(w, v) be the function which results from /(x, y) by the substitution. Then the rule for the differentiation of composite functions gives dm dm d<f> dot dd/ ss — — ■ -^ ^f du dx du dy du dm dm dtf> dm dtp dv dx dv dy dv whence we may find dm/dx and dm/dy\ for, if the determinant l){<^, \l/)/D(u, v) vanished, the change of variables performed wuuld have no meaning. Hence we obtain the equations (25) dm dm dm dx du du dm ^ dm . ^ dm cy cu dv where A^ Bj C, D are determinate functions of u and v; and these formulae solve the problem for derivatives of the first order. They show that the derivative of a function with respect to x is the sum of the two products formed by multiplying the two derivatives with respect to u and v by A and B, respectively. The derivative with respect to y is obtained in like manner, using C and D instead of A and B, respectively. In order to calculate the second derivatives we need only apply to the first derivatives the rule expressed by the preced- ing formulae ; doing so, we find d^m d (dm\ ^ ( A ^^ _t_ c>^*^\ dx^ dx\dx/ dx\ du dv) d I dm d m\ d I dm dm\ du\ du dv I dv\ du dv) or, performing the operations indicated, dx^ \ du^ du dv du du du dv ) \ dudv dv^ dv du dv dv) and we could find d*m/dxdy, d*m/dy^ and the following derivatives in like manner. In all differentiations which are to be carried out we need only replace the operations d/dx and d/dyhy the operations c? a d d du dv du dv 72 FUNCTIONAL RELATIONS [II, §38 respectively. Hence everything depends upon the calculation of the ooefficients A, By C, D. Example I. Let us consider the equation /90\ o — - + 2 — — - + c -— = 0, where the coefficient* a, 6, c are constants ; and let us try to reduce this equa- tion to as simple a form as possible. We observe first that if a = c = 0, it would be superfluous to try to simplify the equation. We may then suppose that c, for example, does not vanish. Let us take two new independent variables u and 0, defined by the equations M = X 4- ay, v = z-\- py, where a and /3 are constants. Then we have dx ' du du ~ du dv d(o du , ^du and hence, in this case, A = B-\, C = a, D = p. The general formulae then fdve S^u _d^u ^ d-2u d^u a»2 "" du'^ dudv dv^ ^u ^20,,, , ^, d'^u , d'^u dxdy du"^ ^ 'dudv dv^ dy^ aw* "^dudv dv^ and the given equation becomes (a + 26a + ca«)^ + 2 [a + 6(a + /3) + ca/3]£^ + (a -i- 26^ + c^)^ = 0. It remains to distinguish several cases. First case. Let l^ - ac>0. Taking for a and /3 the two roots of the equation a + 2 6r + cr* = 0, the given equation takes the simple form Since this may be written = 0. dudv dv\du/ we Me that du/du must be a function of the single variable, u, 8ay/(u). Let P(u) denote a function of u such that F'(u) =/(u). Then, since the derivative of •» — F(u) with respect to u is zero, this difference must be independent of m, and, accordingly, u = F{u) -f ♦ (v). The converse is apparent. Returning to the varlablea x and y, it follows that all the functions u which satisfy the equation (90) are of the form w = F(x + ai/) + «l>(x + /3y), Il,§a«J TRANSFORMATIONS 78 where F and ^ are arbitrary functioua. For example, the geiMral Inlegral of the equatiun — = a* — f which occurs iu the theory of the stretched string, Is «=/(« + «y) + (« - ay). Second case. Let b^ — ac = 0. Taking a equal to the double root of the equa- tion a + 26r + cr3 = 0, and /3 some other number, the coefficient of d^ta/cutv becomes zero, for it is equal to a + 6a + /3 (6 + ca). Hence the given equation retluces to d^u/tv'^ = 0. It is evident that « must be a linear function of v, u = vf(u) + (w), where /(u) and (u) are arbitrary functions. Returning to the variables z and y, the expression for w becomes « = (X + /3y)/(x + ay) + 0(x + ay), which may be written w = [x + «y + 09 - «)y]/(-c + ay) 4- 0(x + ay), or, finally, « = yF{x + ay) + «f>(x + ay). Third case. If 6=* - ac < 0, the preceding transformation cannot be applied without tlie introduction of imaginary variables. The quantities a and /3 may then be determined by the equations o + 2 6a-fca« = a + 2 6/3 + c/3», a + 6(a + /3) + ca/3 = 0, which give 26 ^ 26«-ac a + p = , a/5= c c* The equation of the second degree, . 26 ^ 26«-ac ^ r« + — r4- — = 0, c c* whose roots are a and /3, has, in fact, real roots. The given equation then becomes Aw = — - + — - = 0. au« at* This equation Aw = 0, which is known as Laplace's Equation, is of fundamental importance iu many branches of mathematics and mathematical physics. Example II. Let us see what form the preceding equation assumes when we set X = /} COS0, y = p sin0. For the first derivatives we find dw dw ^ . 5« , ^ — = —0080 + — sin 0, dp dx cV - = --(,»ln«+-,co.#. 74 FUNCTIONAL RELATIONS [II, §39 or» solving for a«*/ac and e«/ay, Su du 8in0 dbt — = CO80 » dx dp p d<f> dta du COS0 d<a — =8in0 1 — • dy dp p o<p ?i = CO.*- (co.* -- — -)-— ^ (cos* ^ - — -j a«« 8in«0 52« 2 8in0co80 d'^u 2,&m<t>cos<t> d(a sm^^ 8fa> and the expreasion for d^u/dy^iB analogous to this. Adding the two, we find ^"^ay2~ap^"^p2^ pap' 39. Another method. The preceding method is the most practical when the function whose partial derivatives are sought is unknown. But in certain cases it is more advantageous to use the following method. Let z =f(xj ^) be a function of the two independent variables x and y. If x, y, and z are supposed expressed in terms of two aux- iliary variables u and v, the total differentials dx, dy, dz satisfy the relation which is equivalent to the two distinct equations du dx du dy du ^ = £/*?^ + ?/?i^, dv dx dv dy dv whence df/dx and df/dy may be found as functions of u, v, dz/duy dg/dvy as in the preceding method. But to find the succeeding derivatives we will continue to apply the same rule. Thus, to find ^f/dz* and ^fldxdy, we start with the identity <i)' iff a»/ which is equivalent to the two equations dx'^ du dx dy du II, §39] TRANSFORMATIONS 76 (^)_ ayax ay dy dv do^ bv dx dy dv where it is supposed that dfjdx has been replaced by its value cal- culated above. Likewise, we should find the values of d^f/dx dy and d*f/dy^ by starting with the identity {©■ ay ^^ ay ^ dx dy dy* The work may be checked by the fact that the two values of d*f/dzdy found must agree. Derivatives of higher order may be calculated in like manner. Apjylication to surfaces. The preceding method is used in the study of surfaces. Suppose that the coordinates of a point of a surface ^' are given as functions of two variable parameters u and v by means of the formulae (27) X = /(m, v), y = <\> (w, v), z = ^(u,v). The equation of the surface may be found by eliminating the vari- ables u and V between the three equations (27); but we may also study the properties of the surface S directly from these equations themselves, without carrying out the elimination, which might be practically impossible. It should be noticed that the three Jacobians D{f.<\>) D{4>^^) D(f, «A) D(uy v) ' D(Uy v) * D{u, V) cannot all vanish identically, for then the elimination of u and v would lead to two distinct relations between x, y, z, and the point whose coordinates are {x, y, z) would map out a curve, and not a sur- face. Let us suppose, for definiteness, that the first of these does not vanish : /)(/, <fi)/D(u, r) ^ 0. Then the first two of equations (27) may be solved for u and u, and the substitution of these values in the third would give the equation of the surface in the form z = F(Xj y). In order to study this surface in the neighborhood of a point we need to know the partial derivatives p, q, r, s,t, • • of this function F(j*, y) in terms of the parameters u and v. The first derivatives p and q are given by the equation dz = pdx + q dyj which is equivalent to the two equations 75 FUNCTIONAL RELATIONS [n,§40 (28) dd, df. d4> from which p and q may be found. The equation of the tangent plane is found by substituting these values of i? and s' in the equation Z-z=p(X-x)-¥qiY-y)y and doing so we find the equation The equations (28) have a geometrical meaning which is easily remembered. They express the fact that the tangent plane to the surface contains the tangents to those two curves on the surface which are obtained by keeping v constant while u varies, and vice versa* Having found i? and q, p =/i(m, v), q =A(u, v), we may proceed to find r, «, t by means of the equations dp = rdx + sd]/j dq = sdx -^ tdy, each of which is equivalent to two equations ; and so forth. 40. Problem IV. To every relation between x, y, z there corresponds hy meant of the equations (30) X =f{u, v, w), y = <f> (u, V, w), z = if;(u, v, w), a new relation between w, v, w. It is required to express the partial derivatives of z with respect to the variables x and y in terms of u, v, Wj and the partial derivatives of w with respect to the variables u and v. This problem can be made to depend upon the preceding. For, if we suppose that w has been replaced in the formulae (30) by a function of u and v, we have a;, y, z expressed as functions of the •Th« «qamtion of the tangent plane may also be found directly. Every curve on tb* wutmM is defined by a relation between u and v, say v = n (u) ; and the equations dl tlM toogent to ibis curve are Y- tu tv du dv du 8v Th* idiiuliintloii iif lV(u\ jM«di to the equation (29) of the tangent plane. II, §41] TRANSFORMATIONS 77 two paxameters u and v; and we need only follow the preceding method, considering f,<f>yilfas composite functions of u and v, and t^; as an auxiliary function of u and v. In order to calculate the first derivatives p and q, for instance, we have the two equations ^.^^^ (^,^^\, (^i^^\ du dw du \du dw du ) ^\du dw duj* diff dw dv \dv dw dv / ^\dv dw dv/ The succeeding derivatives may be calculated in a similar manner. In geometrical language the above problem may be stated as fol- lows : To every point m of space, whose coordinates are (x, y, «), there corresponds, by a given construction, another point 3/, whose coordinates are A", Y, Z. When the point m maps out a surface 5, the point M maps out another surface 2, whose properties it is pro- posed to deduce from those of the given surface .S'. The formulae which define the transformation are of the form Let ^ = /(«» y» «)» Y^^ («, y, 2), Z = ^ (x, y, «). be the equations of the two surfaces S and 2, respectively. The problem is to express the partial derivatives P, Q, Rj S^T, -" of the function *(A', Y) in terms of x, y, z and the partial derivatives Pi ?> ^i s, t, • ■ of the function F(x, y). But this is precisely the above problem, except for the notation. The first "derivatives P and Q depend only on x, y, z, p, q; and hence the transformation carries tangent surfaces into tangent sur- faces. But this is not the most general transformation which enjoys this property, as we shall see in the following example. 41. Legendre's transformation. Let z =/(x, y) be the equation of a surface .S', and let any point m (x, y, z) of this surface be carried into a point 3/, whose coordinates are A', 1', Z, by the transformation X=p, F = 7, Z ^px-^qy — z. Let Z = <I>(A', Y) be the equation of the surface 2 described by the point M. If we imagine z, p, q replaced by /, df/dxy df/dy^ respec* tively, we have the three co6rdinates of the point 3/ expressed as functions of the two independent variables x and y. 78 FUNCTIONAL RELATIONS [II, §41 Lot P, Q, R, Sy T denote the partial derivatives of the function ♦(X, Y). Then the relation rfZ= PdX-\- QdY becomes , _ pdx-{-qdi/ + xdp + ydq — dz = Pdp-\'Qdq, or xdp-\-ydq = Pdp+ Qdq. Let U8 suppose that j9 and q, for the surface S, are not functions of each other, in which case there exists no identity of the form \dp-\-fi.dq — Oj unless X = /i = 0. Then, from the preceding equation, it follows that p = x, Q^y- In order to find R, S, T we may start with the analogous relations dP = RdX-{- SdY, dQ = SdX-{-TdY, which, when X, F, P, Q are replaced by their values, become dx = R(rdx -\- sdy) + S{sdx + tdy), dy = S (rdx + s dy) -t T(s dx + tdy)\ whence Rr^-Ss=l, Rs + St = Oy Sr-{-Ts = Oy Ss-\-Tt = ly and consequently ^^^iiZTi^' ^^H^^r?' ^=;^zr^- From the preceding formulae we find, conversely, x = P, y=Q, z = PX+QY-Z, p = X, q = Y, T - S R t = •— -n -:;> RT-S^ RT-S^ RT-S^ which proves that the transformation is involutory. Moreover, it is a contact transformation, since X, Y, Z, P, Q depend only on x, y» «» Pt 9' These properties become self-explanatory, if we notice that the formulae define a transformation of reciprocal polars with respect to the paraboloid NaU. The expressions for R^ S, T become infinite, if the relation i< — *• = holds at every point of the surface S. In this case the point M describes a curve, and not a surface, for we have II, § 42] TRANSFORMATIONS 79 D(x, y) D(x, y) and likewise DiX,Z) D(p,px + qy-z) D(x,y) D(x,y) '^^"^ '^-'^- This is precisely the case which we had not considered. 42. Ampere's transfonnatioD. Retaining the notation of the preceding article, let lis consider the transformation X = X, r = g, Z-qy-z, The relation dZ-TdX-^ QdY becomes qdy + ydq - dz = Pdx + Qdq, or ydq —pdx = Pdx-\- Qdq. Hence and conversely we find x = X, y=Q, z=QT-Z, p = - P, q = T. It follows that this transformation also is an involutory contact transformation. The relation dP = RdX-{-8dT next becomes -rdx -8dy = Rdz-\- 8{8dx -{-tdy)', that is, B + 5« = - r, 8t= - 8, whence t t Starting with the relation dQ = SdX + TriF, we find, in like manner, t As an application of these formulae, let us try to find all the functions /(z, y) which satisfy the equation rt — s^ = 0. Let S be the surface represented by the equation z =/(x, y), 2 the transtonned surface, and Z = *(X, Y) the equation of 2. From the formulas for /2 it is clear that we must have and 4> must be a linear function of X : z = X0(r) + ^(r), where and \f/ are arbitrary functions of T. It follows th»t P = <f>{Y), Q = X0'(r) + v^'(F); FUNCTIONAL RELATIONS [U, §43 oonrersely, the coordinates (x, y, z) of a point of the surface 8 are given M fonctioDB of the two variables X and Y by the formulae « = X. y = x^'(r) + rm, z = r[X0'(r) + \^'(F)] - x<f>{T) - ^ (F). TIm eqiiAtion of the surface may be obtained by eliminating X and Y ; or, what amounts to the Sftine thing, by eliminating a between the equations z = ay-x<p{a)-yl/{a), = y -X <p'{a) - yp^{a). The first of these equations represents a moving plane which depends upon the parameter a, while the second is found by differentiating the first with respect to this parameter. The surfaces defined by the two equations are the so-called dtMJopabU surfaces, which we shall study later. 4S. The potential equation in curvilinear coordinates. The calculation to which a change of variable leads may be simplified in very many cases by various derioes. We shall take as an example the potential equation in orthogonal earviUnear coordinates.* Let F (X, y, z) = p, Fi{x, y, z) = pi, ■P2(x, y, Z) = p2, be the equations of three families of surfaces which form a triply orthogonal system, such that any two surfaces belonging to two different families intersect at right angles. Solving these equations for x, y, z as functions of the parame- iers p, ^1, ps, we obtain equations of the form « = <t> {p, Pu Pi)i (SI) {« = 0>, Pi, Pih y = 0i(/>, pi, P2), 2 = 02(/), Pi, Pa); and we may take p, pi , ps as a system of orthogonal curvilinear coordinates. Since the three given surfaces are orthogonal, the tangents to their curves of interseotion must form a trirectangular trihedron. It follows that the equations most be aatisfled where the symbol S indicates that we are to replace by 0i, then by ^, and add. These conditions for orthogonalism may be written in the loUowing form, which is equivalent to the above : m 5p£pi_^apapiap£pi_ dx dx dy dy dz dz ~ ' KOZ ax dx dx dtg^Ul^\T'. JJl*'^'*'^*' cur^ign^B. See also Bertrand, TrcdU de CcUcul u. § 4;ij TKA2sbFOUM Allocs 81 Let us then see what form the potential equation in the variables />, pi, ps . First of all, we find dx dp dx dpi dx dpt dx and then ^V _ !^V (Spy , 2 g*^ ^P dpi dV ^p dx^ d(^\dxj dpdpi dx dx dp d^ dVd»pi ^ ^/apiy ^ 2^F_ apt ap, ^ ar apj \dx/ dpi dpt dx dx dpi .^(dptV dfi \dx) dpdpt dx dx dp% dx^ Adding the three analogous equations, the terms containing derivatives of the second order like c^ V / dp cpi fall out, by reason of the relations (33), and we have (34) axa cy^ dz^ dp^ cp\ dp\ + Aa(p) ^' + A,(pi) ^' + A,(p,) 1^, ap api dpt where Ai and Aa denote Lami*B differential parameters : ay2 cz* The differential parameters of the first order Ai (p), Ai (pi), Ai (ps) are easily calculated. From the equations (31) we have a0 dp a0 dp\ a0 apa ap ax api ax aps ax a0i dp^ d<t>i dpi a 01 a Pa dp dx dpi ex dpt dx d4>t dp a0t api a^j ap» ap a* api ax apj ax whence, multiplying by a a 01 a 01 ap dp dp dp dx « respectively, and adding, we find d<f> dp m-<-sy<^} Then, calculating dp/c'y and dp/dz in like manner, it is easy to see that 1 W <%)■<£)■ -777^ m'^fY^m Y S2 FUNCTIONAL RELATIONS [n,§43 Let OB now Mi "sm- -'S{^r- -sm where the symbol ^ Indicatee, as before, that we are to replace by 0i, then by ^, and add. Then the preceding equation and the two analogous equations may be written AiO>) = -^' AiO>i) = — , Ai(p2) = — • H *^*' Hi '^"' H. Lam4 obtained the expressions for Aj (p), A2 (pi), A2 (pa) as functions of p, pi, ^ by a rather long calculation, which we may condense in the following form. In the idenUty (34) AiF=-— - + — — y + — -— y + Aap)— + A2(pi) — - + A2(P2) — > H ap* Hi df^ Hi ap2 ap api ap2 let us set succeasively F = x, F = y, F = z. This gives the three equations 1 a»0 1 a«0 1 av . . , V S0 a^ a0 _ H V ^^^ ^:s^ ""^"^'^^ +^^^^>^, +^^^^^>^2 ='' n- TT + W" TT + ^ VF + ^2 (p) -— • + A2(pi) -— + A2 (P2) -^- = 0, B ap* Hi dpi Ht dp2 ^P dpi dpi 1 a«^ 1 a»0, 1 a202 . . , . a02 , . , . a02 , . . . a02 ^ w HT "•■ IF TF + w Ti" + ^a(^) ^- + ^2(pi) ^— + A2(p2) ^ = 0, H dpi* Hi dp^ Hi dpi dp dpi dpi which we need only solve for Aj (p), A2 (pi), A2 (pi). For instance, multiplying by d^/dpy d^/dp^ dipi/dpy respectively, and adding, we find A-Z-^Wj. ^ e^^^^x ^ nd<t>d^<t>, 1 nd<f>d2<f>_ MbvMTer, we have oa0 a20 lag ^ ap ap2 ~ 2 ap * and differentiating the first of equations (32) with respect to pi, we find oa0a«0___ nd<p 8^<f> _ 1 dHi *^ dp^ ^ dpidpdpi~ 2 dp ' In like manner we have *^ ap dpi " 2 ap ' MdMonqontiy A.(#)---L?f 4.-_^_«^t . 1 dHt_ 1 ar / g \n iHtf^^'^nni'W^^Wm'di'-'^VpV'^KWHjJ' II. EX8.] Setting thiB formula becomes "=r«' £X£RCISES ^' = M' 88 ^• = i^' and in like manner we find Hence the formula (84) finally becomes ap2 ap \ ^AiV 2/> . .op^r a /, A, \ar" (86) ax« ai/a az2 or, in condensed form, Ldp \hihi dp I dpi \hht dpi I dpi \hhy dpi/J Let us apply this formula to polar coordinates. The formulaB of transforma- tion are X = psin(?cos0, y = psindeiiKp, z = pcoB$, where 6 and <f> replace pi and po, and the coeflRcients h. h^. h^ have the following Talues: A=l, hi=\ /i, = -i_. p p sin 9 Hence the general formula becomes AaF= — (p^sin^ ) + — (smtf — ) + — ( I h P^sin e\_dp\ dp dd\ de d<f> Vsin e d4> J or, expanding, AaF dp^ p2 d0^ p«8ln«tf d<t^ ' p dp a*F 2 aF cottf aF which is susceptible of direct verification. EXERCISES 1. Setting u = x« + y2 + 2«, tj = X 4- y + z, w = xy + yz + «x, the functional determinant D(u, t>, w)/D(x, y. z^ vanishes identically. Find the relation which exists between u, o, to. OtMTolixe the problem. OA FUNCTIONAL RELATIONS [H. Em. VT^ S. Using the notation Xi = cos 01, Xt = 8in0icos02, Xt = sin 01 sin ^a cos 0$, » x« = sin 01 sin 02 • • • sin 0„ _ 1 cos 0„, •how that Djxu x«« -t »«) = ( _ i)n sin" 01 sin" - 1 02 sin" - 208 . • • sin2 0»_i sin 0„. D(0i.0sf •,0») 4. Prove directly that the function z = F(x, y) defined by the two equations z = ax-h yf{a) + (a), = X + yria) + 0'(a), where a is an auxiliary variable, satisfies the equation rt - s"^ = 0, where f{a) Aod 0(<r) are arbitrary functions. 5. Show in like manner that any implicit function z = F{x, y) defined by an equation of the form y =X0(z) + t/'(z), where ^ (z) and ^ (z) are arbitrary functions, satisfies the equation rg2 _ 2i)5S + tp2 = 0. 6. Prove that the function z = F{x, y) defined by the two equations z 0'(a) = [y - (or)] 2, (X + a) 0'(a) = y - (or), where a la an auxiliary variable and {a) an arbitrary function, satisfies the equation pq = t. 7. Prove that the function « = F(x, y) defined by the two equations [t - 0(a)]« = x«(y« - a«), [2-0 (a)] 0'(a) = ax^ ■■tliflei in like manner the equation pq = xy. fl». Lafraage's formula. Let y be an implicit function of the two variables X Mid a, defined by the relation y = or + x0(y); and let u =f{y) be any func- tion of y whatever. Show that, In general, ftp* do— iL (fa J [Laplace.] II, ExB.J EXERCISES 85 Note. The proof Ib based upon the two formal» where it !■ any function of y whatever, and F(u) is an arbitrary function of 11. It la shown that if the formula hulds for any value of n, it must hold for the value n + 1. Setting z = 0, y reduces to a and u to /(a); and the nth derivative of u with respect to x becomes (a=.^.[*<''>"^'<^>]- 9. If X =/(u, 0), y = (u, v) are two functions which satisfy the equations show that the following equation is satisfied identically : 10. If the function F(x, y, z) satisfies the equation show that the function I exa aya dz^ r \ r^ r^ r*/ satisfies the same equation, where k is a constant and r^ = x^ + y^ + z>. [Lord Kelvin.] 11. If F(x, y, 2) and Fi(z, y, z) are two solutions of the eijuation AjF = 0, show that the function U = F(x, y, 2) + (x« + y2 + z^) Fi(x, y, «) satisfies the equation 12. What form does the equation (X - x»)y"+ (1 - 3x2)y'- xy = assume when we make the transformation x = Vl —t^? 13. What form does the equation ?f + 2xy«?^ 4- 2(y - y«)?^ + x^y^z = dx^ dx cy assume when we make the transformation x = ut>, y = l/o ? 14*. I^t (xi, xt, • • • , Xn ; ui, Us, • • •, Um) be a function of the 2n independent variables xi, x^, • • • , x^, ui, us, •••,!««, homogeneous and of the second degree with respect to the variables Wi, mj, • • •, u„. If we set M FUNCTIONAL RELATIONS [U, Exs. and then take pi , Pj, • • • , P» as independent variables in the place of Wi, 1*2, • • • , Wn, the fancUoD ^ goes over into a function of the form D«riTe the formuls : ., ^^ dpk ^k OXk 16. Let JV be the point of intersection of a fixed plane P with the normal MI^ erected at any point Jf of a given surface S. Lay off on the perpendicular to the plane P at the point N a length Nm = NM. Find the tangent plane to the surface described by the point m, as 3f describes the surface S. The preceding transformation is a contact transformation. Study the inverse transformation. 16. Starting from each point of a given surface S, lay off on the normal to the surface a constant length I. Find the tangent plane to the surface S {tlie parallel aurface) which is the locus of the end points. Solve the analogous problem for a plane curve. 17*. Given a surface S and a fixed point O ; join the point O to any point M of the surface 8, and pass a plane OMN through OM and the normal MN to the surface S at the point Jf. In this plane OMN draw through the point a per- pendicular to the line OM, and lay off on it a length OP = OM. The point P describes a surface S, which is called the apsidal surface to the given surface S. Find the tangent plane to this surface. The transformation is a contact transformation, and the relation between the surfaces 8 and 2) is a reciprocal one. When the given surface S is an ellipsoid and the point is its center, the surface Z is FresnePs wave surface. 18*. Halphen'8 differential invariants. Show that the differential equation \dxV dx6 d«2 dx« dx* \dx»/ remaina unchanged when the variables cc, y undergo any projective transfor- mation (S 87). 19. If in the expression Pdx -f Qdy -\- Rdz, where P, Q, R are any functions of X, y, z, we set x=/(u, t), u>), 2/ = (u, t), ry) , z = rp {u, v, w) , whan tt, V| 10 are new variables, it goes over into an expression of the form Pidu+ Qidt» + Bidiy, Pit Qi* J^i are functions of u, t), w. Show that the following equation is identioally: D{u, t>, to) II. Ex«.] EXKRCI8E8 87 when ''■'■('«'-S*C-S*«(^-S). *-''.(l?-^')**('^'-i?)*»'(¥-5?> 20*. Bilinear covariants. Let 6^ be a linear differential form: Gd = Xidxi + Xtdxt + • • • + JT.dx,, where Xi, Xt, • • •, X^ are functions of the n variables Zi, Zf, • • • , z.. Let us consider the expression n n where a* __ ^-^< _ ^^k dXk dXi and where there are two systems of differentials, d and 8. If we make any transformation z. = 0.(yi, ya, •••, 2/n), (t = l, 2, ..-, n), the expression 6^ goes over into an expression of the same form Qd=Tidyi+--^Yndyn, whuru I'l, I'a, •• • , F„ are functions of yi, yj, • • • , y„. Let us also aei , 5 r, a Ft <'-«* = -I ; — dyt dvi and t it Show that 77 = 77', identically, provided that we replace dx, and tojt, respec- tively, by the expressions The expression £7 is called a hUinear covaHard of 9^. 21*. Beltrami's differential parameters. If in a given expression of the form ^dx2 + 2Fdxdy ■\- Gdy^, where E, F, G are functions of the variables x and y, we make a transformation '•^ =/(«i f), y = <f>{Uy t), we obtain an expression of the same form: Ex du^ + 2 Fi du dv 4- (?i di»«. 88 FUNXTIONAL RELATIONS [II, Exa J^u Fit Oi %Te tunctiouB of u and v. "Let d{x, y) be any function of the imriablM z and y, and ^i (u, o) the transformed function. Then we have, iden- tieaUy, \dx/ dx tiy \dvf _ \du / du dv \dv / ^ EG- F^ E^G^-Fl d0 „ de\ /„ de „dd Vmo^F* ^ [ ax £^1-4- 1 , g I gy dx \ \ ^EO - F^ I ^EG-F^ ^\ ^EG - F^ I I du dv ] . 1 d I dv du \ ^E, 6?, - F? / 22. Schwanijui. Setting y = (ax + 6) / (ex + 6), where x is a function of t and a, 6, e, d are arbitrary constants, show that the relation 7f 2 2 Vx7 2/' 2 Uv it Identically satisfied, where x', x'\ xf'\ y\ y", y"' denote the derivatives with ratpect to the variable t. 23*. Let u and t» be any two functions of the two independent variables x and y, and let us set a"u + 6"» + c" ' ~ a''M + 6''t) + c"* where a, 6, c, • • • , c" are constanta. Prove the formulae : ^g»_^aM <f^V dV d^V dU dx^ dx ax2 dx 'd^ 'dx ~ ax^" 'dx (Wi «) (Cr, V) ^u dv dx* , ... .„ (M,tj) go _ g« au /ap d^u du d^v\ dy dx'^ dy \dx dxdy 'dx dx dy) (M,tj) ^i^_^^_U /dV ^V_ _ dU d*V \ _ ^ dy dx^ dy \dx dx dy dx dx dy) the analogoue formulae obtained by interchanging x and y, where dx dy dy dx' ^ ' ' dx dy dx dy' [OouBSAT and PainlkvI:, Comptes rendus, 1887.] CHAPTER III TAYLOR'S SERIES ELEMENTARY APPLICATIONS MAXIMA AND MINUdA I. TAYLOR'S SERIES WITH A REMAINDER TAYLOR'S SERIES 44. Taylor's series with a remainder. In elementary texts on the Calculus it is shown that, if J\x) is an integral polynomial of degree 7i, the following formula holds for all values of a and h: (1) /(« + A) =/(«) + J /'(a) + ^ r{a) + . . . + ^-^/<.>(a). This development stops of itself, since all the derivatives past the (n -h l)th vanish. If we try to apply this formula to a function f{x) which is not a polynomial, the second member contains an infinite number of terms. In order to find the proper value to assign to this development, we will first try to find an expression for the difference /(« + h) -f(a) - J f{a) - ^ f'ia) _^/(.)(„), with the hypotheses that the function /(«), together with its first n derivatives /'(x), /"(«)) • • • , /^"X*)? ^^ continuous when x lies in the interval (a, a 4- h), and that f^*\x) itself possesses a derivative y(« + i)^^^ in the same interval. The numbers a and a -{• h being given, let us set (2) where p is any positive integer, and where P is a number which is defined by this equation itself. L<»t us then consider the auxiliary function 89 90 TAYLOR'S SERIES [UI,§44 ^(x) =/(a + A) ^/(x) - ^±^/'(.) - i^^3^^ 1.2 -n ^ ^ \ 1.2 ..n.j9 ^' It is dear from equation (2), which defines the number P, that ^(a) = 0, <^(a + A) = 0; and it results from the hypotheses regarding f(x) that the func- tion ^(x) possesses a derivative. throughout the interval (a, a + h). Hence, by RoUe's theorem, the equation <^'(a;) = must have a root a '\- $h which lies in that interval, where d is a positive number which lies between zero and unity. The value of <^'(x), after some easy reductions, turns out to be ♦'("'> = ^°t.2~«''"' [^ - (« + A - xy-'^-^r-^'^x)-]. The first factor (a -\- h — xy ~ ^ cannot vanish for any value of x other than a -f A. Hence we must have p = A— " + » (1 - d)«-i'-^ »/(»+») (a + Oh), where < d< 1 ; whence, substituting this value for P in equation (2), we find (3) /(a + *)=/(a) + ^/'(a)+ j^/"(a) + . . . + _^/(»)(„)+fl,, where ^-= 1^2...;.^ /"--'(a + OA). We shall call this formula Taylor's series with a remainder, and the last term or R^ the remainder. This remainder depends upon the poeitive integer />, which we have left undetermined. In practice, about the only values which are ever given to p are p = n + 1 and /» ■» L Setting p = n 4- 1, we find the following expression for the remainder, which is due to Lagrange : ^•° l .2.^(1 + 1/ '""^('' + ^">' ■ettixig/>a.l, we find Ill, § 44] TAYLOR'S SERIKS WITH A REMAINDER 91 an expression for the remainder which is due to Cauchy. It is clear, moreover, that the number will not be the same, in general, in these two special formulae. If we assume further that /<""*■ *^(x) is continuous when x = a, the remainder may be written in the form where c approaches zero with h. Let us consider, for definiteness, Lagrange's form. If, in the gen- eral formula (3), n be taken equal to 2, 3, 4, • • • , successively, we get a succession of distinct formula which give closer and closer approximations for /(a -|- h) for small values of A. Thus for n = 1 we tind /(« + A) =/(a) + j/'(a) + j^/" (a + tfA) , which shows that the difference /(„ + A) _/(„) - ^/'(a) is an infinitesiinal of at least the second order with respect to A, provided that /" is finite near x = a. Likewise, the difference is an infinitesimal of the third order ; and, in general, the expression /(« + A) -/(«) - j/'(«) ^/*"*(«) is an infinitesimal of order n -\- 1. But, in order to have an exact idea of the approximation obtained by neglecting /?, we need to know an upper limit of this remainder. Let us denote by M* an upper limit of the absolute value of /^*"*'*>(x) in the neighborhood of X = a, say in the interval (a — 17, a H- 1;). Then we evidently have p |< IUlL M provided that | A | < 17. • That is, M > !/(•• + i){x) | when | x - a | < i». The expression *' the upper limit," defined in § tiS, must be carefully distinguished from the expression " an upper limit," which is used here to denote a number greater than or equal to the absolute Talue of the function at any point in a certain interval. In this paragraph and in the next /<■ + »)(») is supposed to have an upper limit near x = a. — Tbaks. 92 TAYLOR'S SERIES [III, §45 45. ApplicAtioii to curves. This result may be interpreted geomet- rically. Suppose that we wished to study a curve C, whose equa- tion is y =/(«), in the neighborhood of a point Aj whose abscissa is a. Let us consider at the same time an auxiliary curve C, whose equation is K=/(a) + ^/'(«) + ^/»+- + fe^/-W. A line x = a -{- h^ parallel to the axis of y, meets these two curves in two points M and 3/', which are near A. The difference of their ordinates, by the general formula, is equal to This difference is an infinitesimal of order not less than w + 1 ; and consequently, restricting ourselves to a small interval (a — rj, a -\- ?;), the curve C sensibly coincides with the curve C". By taking larger and larger values of n we may obtain in this way curves which differ less and less from the given curve C; and this gives us a more and more exact idea of the appearance of the curve near the point A. Let us first set n = 1. Then the curve C is the tangent to the curve C at the point A : r=/(a) + (x-a)/'(a); and the difference between the ordinates of the points M and M' of the curve and its tangent, respectively, which have the same abtdisa a + A, is Ut us suppose that /"(a) =^ 0, which is the case in general. The pMceding formula may be written in the form where c approaches zero with h. Since /"(a) =?t 0, a positive num- ber iy can be found such that |c| < |/"(a) |, when h lies between - rj and -f. If. For such values of h the quantity /"(a) + « will have the aame sign as /"(a), and hence y - r will also have the same iigli ai/"(a). Iff "(a) is positive, the ordinate y of the curve is 111,546] TAYLOR'S SERIFIS WITH A REMAINDER 98 greater than the ordinate Y of the tangent, whatever the sign of h ; and the curve (' lies wholly above the tangent, near the point A, On the other hand, if /"(a) is negative, y is less than K, and the curve lies entirely below the tangent, near the point of tangenoy. If /"(a) = 0, let /^'''(a) be the first succeeding derivative which does not vanish for x =i a. Then we have, as before, if f^'\x) is continuous when x = a, y- I' =177^ [/""(«) + «]; and it can be shown, as above, that in a sufficiently small interval (a — i;, a -I- »;) the difference y — Y has the same sign as the product hpf(p^(^a). When j^ is even, this difference does not change sign with hf and the curve lies entirely on the same side of the tangent, near the point of tangency. But if p be odd, the difference y — Y changes sign with A, and the curve C crosses its tangent at the point of tangency. In the latter case the point A is called a point of inflection ; it occurs, for example, if f"\a) ^ 0. Let us now take n = 2. The curve C is in this case a parabola : r=f(a) + (X - o)/'(a) + ^^fj^rW, whose axis is parallel to the axis of y; and the difference of the ordinates is y- 1- =1:1:3 [/"'(«) + «]• If /'"(a) does not vanish, y — Y has the same sign as h*f"'(a) for sufficiently small values of h, and the curve C crosses the parabola C at the point A. This parabola is called the oscuUUory parabola to the curve C ; for, of the parabolas of the family Y = 7nx^ -\- nx -^ Pf this one comes nearest to coincidence with the curve C near the point .4 (see § 213). 46. General method of development. The formula (3) affords a method for the development of the infinitesimal f(a -\- h) —/(a) according to ascending powers of h. But, still more generally, let X be a principal infinitesimal, which, to avoid any ambiguity, we 94 TAYLOR'S SERIES [III, §46 will suppose positive ; and let y be another infinitesimal of the form (4) y = ^ix^ + A^T** 4- . . • 4- a^-'C^p + «), where n„ «,,.••, n, are ascending positive numbers, not necessarily integers, .4i, ^4,, •••, A^ are constants different from zero, andc is another infinitesimal. The numbers Wi, ^i, n^, A^, •■■ may be cal- culated successively by the following process. First of all, it is clear that n, is equal to the order of the infinitesimal y with respect to x, and that Ai is equal to the limit of the ratio y/x^i when X approaches zero. Next we have y - AiX^i = ui = AiX^'t H h (^p + €)»"', which shows that w, is equal to the order of the infinitesimal Wi, and At to the limit of the ratio Wi/cc"*. A continuation of this process gives the succeeding terms. It is then clear that an infini- tesimal y does not admit of two essentially different developments of the form (4). If the developments have the same number of terms, they coincide; while if one of them has p terms and the other p •\- q terms, the terms of the first occur also in the second. This method applies, in particular, to the development off (a -\- h) —/(a) according to powers of h ; and it is not necessary to have obtained the general expression for the successive derivatives of the func- tion f(x) in advance. On the contrary, this method furnishes us a practical means of calculating the values of the derivatives A«). /».•••• Examples. Let us consider the equation (5) ^i^fV) = Axr •{. By + xy^(x, y) -^ Cx-^' -{-•••+ Dy^ + "■ = 0, where ♦ («, y) is an integral polynomial in x and y, and where the termi not written down consist of two polynomials P(x) and Q(y), which are divisible, respectively, by a;" + ' and y\ The coefficients A Aod B are each supposed to be different from zero. As x approaches wro there is one and only one root of the equation (5) which ap- proaohea rero (§ 20). In order to apply Taylor's series with a remainder to this root, we should have to know the successive deriv- ativee, which could be calculated by means of the general rules. But we may proceed more directly by employing the preceding method. For this purpose we first observe that the principal part Ill, §4<J] TAYLOR'S SERIES WITH A REMAINDER 95 of tlie intinitesimal root is equal to — (^4 /B)x". For if in the equa- tion (5) we make the substitution y H*^)- and then divide by j;", we obtain an equation of the same form (6) { ^\(«» y\) = ^lX^ -f- %i -I- xy^ *i {x, y,) -I- CiX". + '-h- •+ /^//?4-... = 0, which has only one term in yi, namely Hy^' As a* approaches zero the equation (6) possesses an infinitesimal root in y,, and conse- quently the infinitesimal root of the equation (5) has the principal part — (^//?)x'', as stated above. Likewise, the principal part of yx is — (^li/i^)x"t; and we may set y=-|x- + ^-^ + y,jx- + -i, where y, is another infinitesimal whose principal part may be found by making the substitution y. = a-.(-|i + y.j in the equation (6). Continuing in this way, we may obtain for this root y an expres- sion of the form y = arr" -h a,x" + "t + a,x"+*t + *» ^ h («p + e)a;" + "i-»-- "^V, which we may carry out as far as we wish. All the numbers n, n,, n,, •••, n^ are indeed positive integers, as they should be, since we are working under conditions where the general formula (3) is applicable. In fact the development thus obtained is precisely the same as that which we should find by applying Taylor's series with a remainder, where a = and h — x. I^et us consider a second example where the exponents are not necessarily positive integers. I^et us set ^a;*4- 5x^4- Car>-f ••• y = 1 -f J5iX^»-|- Cix>' H 96 TAYLOR'S SERIES [III, §46 where a, ft 7, ••• and ft, yi, • • • are two ascending series of positive nomben, and the coefficient A is not zero. It is clear that the prin- dpal part of y is Ax^j and that we have which is an expression of the same form as the original, and whose principal part is simply the term of least degree in the numerator. It is evident that we might go on to find by the same process as many terms of the development as we wished. Let/(x) be a function which possesses n + 1 successive derivatives. Then replacing a by z in the formula (3), we find /(x + A) =/(x) + ^r{x) + r^r(x) + ■ • • + 7-^ [/<">(x) + e] , wbere c approaches zero with h. Let us suppose, on the other hand, that we had obtained by any process whatever another expression of the same form for /{x + h): /(X + h) =/(x) + h<f>i{x) + h^</>2{x) + • • • + /i" IMX) + 61. Tbete two developments must coincide term by term, and hence the coefficients ^» ^. • • • » 0)» are equal, save for certain numerical factors, to the successive derivatives of f(x) : ^i(x)=/'(x), 0,(x)=-^, ..., 0„(x)- -^^"^^"^^ 1.2 ' ^"'' 1.2...n This remark is sometimes useful in the calculation of the derivatives of certain functions. Suppose, for instance, that we wished to calculate the nth derivative of a function of a function : y = /(u) , where u = (x) . Neglecting the terms of order higher than n with respect to h, we have k = ^(x + A) - 0(x) = J 0'(x) + -^ 0"(x) -f- . • + , j^" 0(«)(x); 1 1.2 1.2- n and Ukewiee neglecting terms of order higher than n with respect to *, 1 I . Z 1 . iS • • • H If in the right-hand side * be replaced by the expression and the reeultlng expresalon arranged according to ascending powers of A, it is erident that the t«rma omitted will not affect the terms in ^, ^a, . . . , A". The III. f 47] TAYLOR'S SERIES WITH A KL^lAlM^tU 97 coefficient of A", for instance, will be equal to the nth deriymtite of /[^(x)] diTided by 1 . 2 • • • n ; and hence we may write where Ai denotes the coefficient of A" in the deyelopment of For greater detail concerning this method, the reader is referred to Hermite's Coura d' Analyse (p. 50). 47. Indeterminate forms.* Let f(x) and i>(x) be two functions which vanish for the same value of the variable x = a. Let us try to find the limit approached by the ratio /(a + h) as h approaches zero. This is merely a special case of the problem of finding the limit approached by the ratio of two infinitesimals The limit in question may be determined immediately if the prin- cipal part of each of the infinitesimals is known, which is the case whenever the formula (3) is applicable to each of the functions f(x) and </> (x) in the neighborhood of the point a. Let us suppose that the first derivative of f(x) which does not vanish for x = a is that of order p, f^*'^{a) ; and that likewise the first derivative of ^ {x) which does not vanish for x = a is that of order y, ^^«^(a). Applying the formula (3) to each of the functions /(x) and ^(x) and dividing, we find /(^ + A) ^ 1.2..-y/n«)-fc where e and «' are two infinitesimals. It is clear from this result that the given ratio increases indefinitely when h approaches zero, if q is greater than i> ; and that it approaches zero if q is less than p, U q =Pf however, the given ratio approaches f^''\<t)/4>^^^(o) as its limit, and this limit is different from zero. Indeterminate forms of this sort are sometimes encountered in finding the tangent to a curve. Let z =/(<), y = 0(e), t = ^(0 • See also $ 7. 98 TAYLOR'S SERIES [m,§48 be the equationa of a curve C in terms of a parameter t The equations of the tan^eiit to this curve at a point M, which corresponds to a value U of the param- «|«r, are, M we saw in § 5, Z-f{to) ^ F-0(<o) ^ Zj-mKM. /'(to) 0'(^o) V''(«o) Thmt equaUona reduce to identities if the three derivatives /(t), 0'(t), \p'{t) aU TaDlsh for t = t©. In order to avoid this difficulty, let us review the reasoning by which we found the equations of the tangent. Let M' be a point of the curve C near to 3f, and let <o + A be the corresponding value of the parameter. Then the equationa of the secant MM' are Z-f{to) ^ r-0(<o) ^ Z-ypjto) /{to + A) -/(to) 0(to + A) - 0(to) ^ {to + h)- ypito) For the sake of generality let us suppose that all the derivatives of order less than p(p> 1) of the functions /(<), <t> (0, ^ (0 vanish for t = to, but that at least one of the derivatives of order p, say /<p) {to), is not zero. Dividing each of the denominators in the preceding equations by hp and applying the general for- mula (8), we may then write these equations in the form X-f{to) ^ T-4>{to) ^ Z-rp{to) f^P^ (to) + e 0('» (to) + e' ^(1') (to) + c" ' where e, «', «" are three infinitesimals. If we now^ let h approach zero, these equations become in the limit X-f{to) ^ r-0(to) ^ Z-V^(to) /(p)(to) 0(i')(to) ^(P)(to) ' In which form all indetermination has disappeared. The points of a curve C where this happens are, in general, singular points where the curve has some peculiarity of form. Thus the plane curve whose equations are X = t2, y = «8 through the origin, and dx/dt = dy /dt = a,t that point. The tangent is the axis of z, and the origin is a cusp of the first kind. 48. Taylor's series. If the sequence of derivatives of the function /(x) is unlimited in the interval (a, a + h), the number n in the formula (3) may be taken as large as we please. If the remainder R^ approa/ihes tero when n increases indefinitely, we are led to write down the following formula : (7)/(« + A) = /(a) + J/'(a)+j^r(a) + ... + j-|--/(-)(„) + ..., which expresses that the series m,$18] TAYLORS SERIES WITH A ROIAINDER 99 is convergent, and that its " sum " ♦ is the quantity /(a 4- A). This formula (7) is Taylor's aeries, properly speaking. But it is not justi- fiable unless we can show that the remainder R^ approaches zero when n is infinite, whereas the general formula (3) assumes only the exist- ence of the first n -f 1 derivatives. Keplacing a by as, the equation (7) may be written in the form /(x + A) = /(x) +*/'(*)+.. + -j-^ /<">(x) +... . Or, again, replacing A by x and setting a = 0, we find the formula (8) /(x) =/(0) + I /'(O) + . . . + j-^ /'•>(0) + . . .. This latter form is often called MaclaurirCs series; but it should be noticed that all these different forms are essentially equivalent. The equation (8) gives the development of a function of x accord- ing to powers of x ; the formula (7) gives the development of a func- tion of h according to powers of A : a simple change of notation is all that is necessary in order to pass from one to the other of these forms. It is only in rather specialized cases that we are able to show that the remainder R^ approaches zero when n increases indefinitely. If, for instance, the absolute value of any derivative whatever is less than a fixed number M when x lies between a and a -f A, it follows, from Lagrange's form for the remainder, that 1 A I " + > l^^-l<-^ ^.2...(n + l) ' an inequality whose right-hand member is the general term of a convergent series.f Such is the case, for instance, for the functions e'j sin Xy cos x. All the derivatives of e' are themselves equal to e*, and have, therefore, the same maximum in the interval con- sidered. In the case of sin x and cos x the absolute values never exceed unity. Hence t/ie formula (7) is applicable to these three functions for all values of a and A. Let us restrict ourselves to the form (8) and apply it first to the function f(x) = e». We find /(0) = 1, /'(0) = 1, •.., /<">(0) = 1, ...; * That is to say, the limit of the sura of the first n terms as n becomes Infinite. For a definition of the meaning of the technical phrase '* the stun of a terits" see § 1.77. — Trans. f The order of choios is a, h, M, n, not a, h, n, M. This is essential to the oon* Terf^enco of the series in question. — Trans. 100 TAYLOR'S SERIES [in,§49 and Qonaequently we have the formula (9) ^=1 + 1 + 172^ ••■^r:2T:^"^**'' which applies to all values, positive or negative, of x. If a is any positive number, we have a' = e'>'«-, and the preceding formula becomes (10) ^ , _^^log£ , {xloga}_^ {xXo^ar <»^=^+~l[ "^ 1.2 ^1.2--7i Let us now take f{x) = sin x. The successive derivatives form a recurrent sequence of four terms cos x, — sin x, — cos a;, sin a; ; and their values for x = form another recurrent sequence 1, 0, — 1, 0. Hence for any positive or negative value of x we have X iC* ^ (11) 8inx = j-j-273 + i 2.3.4.5"*" ■^(~-^)"l.2.3. •(2714-1)'^*"* and, similarly, (12) eo»'' = l-0 + i.2^V4 ~--- + <~^>" l.2.3---2»t + -- Let us return to the general case. The discussion of the remain- der R^ is seldom so easy as in the preceding examples; but the problem is somewhat simplified by the remark that if the remain- der approaches zero the series /(«) + */'(«) + ••■ + jtI^ /<"'(«)+•• • neoessarily converges. In general it is better, before examining /J„ to see whether this series converges. If for the given values of a and h the series diverges, it is useless to carry the discussion further ; we can say at once that R^ does not approach zero when n increases indefinitely. 49. Development of log(l -f x). The function log(l -f «), together with all its derivatives, is continuous provided that x is greater than — 1. The successive derivatives are as follows : lll.nyj TAYLOR'S SERIES WITH A REBiAINOEB lOl /"(') = (ItV.' Let us see for what values of x Maclaurin's formula (8) may be applied to this function. Writing first the series with a remainder, we have, under any circumstances, log(l+x) = f-| + | + -.. + (-l)— •{+/.. The remainder R^ does not approach zero unless the series converges, which it does only for the values of x between — 1 and -f 1, including the upper limit -f 1. When x lies in this interval the remainder may be written in the Cauchy form as follows : "" 1.2 -n (1-hdx)--* ^ ^ (H-ftr)-+» or Let us consider first the case where |x|<l. The first factor x appi caches zero with x, and the second factor (1 ~ 6)/(l -h ftc) is less than unity, whether x be positive or negative, for the numer- ator is always less than the denominator. The last factor remains finite, for it is always less than 1/(1 — \x\). Hence the remainder R^ actually approaches zero when n increases indefinitely. This form of the remainder gives us no information as to what happens when X = 1 ; but if we write the remainder in Lagrange's form, 1 1 ^. = (-1)" n + l(l+$Y-^' it is evident that A', a})proa("hes zero when n increases in(lefinit«'ly. An exanunation of the remainder for x = — 1 would be useless, 102 TAYLOR'S SERIES fill. §49 since the series diverges for that value of x. We have then, when X lies between — 1 and -h 1, the formula (13) log(l + «) = f-f + |'-- + (-l)"-'f + -. This formula still holds when a; = 1, which gives the curious relation (U) log2 = l-i + ^-i+- -t-(-l)-i+-. The formula (13), not holding except when x is less than or equal to unity, cannot be used for the calculation of logarithms of whole numbers. Let us replace a; by — x. The new formula obtained. (13-) log(l-x)=--- X o^ ^ _ _ ^ __ ^ ^ still holds for values of x between — 1 and -|- 1 ; and, subtracting the corresponding sides, we find the formula When X varies from to 1 the rational fraction (1 + a;) / (1 — x) steadily increases from 1 to + oo, and hence we may now easily cal- culate the logarithms of all integers. A still more rapidly con- verging series may be obtained, however, by forming the difference of the logarithms of two consecutive integers. For this purpose let us set H-x N-\-l 1 J ov x = 1-x N. 2iV-l-l Then the preceding formula becomes log(;.+l)-log;v=2[^ + 3-^^X_ + ^-^^ + ...]. an equation whose right-hand member is a series which converges very rapidly, especially for large values of N. Note, l^ lu apply the general formula (3) to the function log (1 + x), setting a = 0, A = z, n = 1, and taking Lagrange's form for the remamder. We find in thiiway log(l -f x) = x- 2(1 + ftc)« 111. §4yj rWlOR'S SKUIKS WITTT V REMAINDER 108 If we uow repliicu x i . ol an uiieger n, this m«y be written ^\ n/ n 2ii« where 9. is a positive number less than nnity. Some interesting conaeqnenoes may be deduced from this equation. 1) The liarmonic series being divergent, the sum inoretses indefinitely with n. But tlie difiference 2,, - log n approaches a finite limit. For, let us write this difference in the form (>-.o«^).g-.o.0.....(l-.o.a±i) + • P I n+ 1 Now 1 /p - log (1 + 1 /p) is the general term of a convergent series, for by the equation above p \ vJ 1p 2p« which shows that this term is smaller than the general term of the convergent series 2(1 / p^). When n increases indeiinitely the expression n \ n/ approaches zero. Hence iho difference under consideration approaches a finite limit, which is called Euler'a coiistant. Its exact value, to twenty places of decimals, U C = 0.67721660490163286060. 2) Consider the expression > +_L, + . .. + _!_. n+1 n+2 n+p where n and p are two positive integers which are to increase indefinitely. Then we may write -(■*;* ■*=f,)-(-i*-;)- l + ^+-+~^ = l0g(»+p)+/Ni^,. 1+^ + - +^ =l0gll + /N„ 104 TAYLOR'S SERIES [ill, §50 where ^+, and ^ approach the same value C when n aiid p increase indefi- nitely. Heooe we have also = log(l+?) + Pn+p - Pn Now the difference p,+p - Pn approaches zero. Hence the sum S approaches DO limit unle« the ratio p/n approaches a limit. If this ratio does approach a limit a, the sum 2 approaches the limit log (1 + a). Setting p = n, for instance, we see that the sum iTTl "^ n + 2 ' " * 2 n approaches the limit log 2. 50. Development of (1 + x)". The function (1 4- x)"" is defined and continuous, and its derivatives all exist and are continuous func- tions of aj, when 1 + « is positive, for any value of m ; for the derivatiyes are of the same form as the given function: /'(x)=m(l +x)''-S /"(ar)=m(m-l)(l+x)'"-2, m — n — 1 /<">(a;) = m(w - 1) • • • (m - 71 4- 1) (1 + ^) /(• + i)(x) = w (w - 1) • • • (m - w) (1 4- «) Applying the general formula (3), we find (l+x). = l+^x + t;^l^x^ + ... and, in order that the remainder R^ should approach zero, it is first of all necessary that the series whose general term is m(m — 1) • • • (m — n + 1) ^ 1.2-..n * should converge. But the ratio of any term to the preceding is m — n 4- 1 which approaches — x as n increases indefinitely. Hence, exclud- ing the case where m is a positive integer, which leads to the ele- mentary binomial theorem, the series in question cannot converge |x|< t. Let us restrict ourselves to the case in which | x { < 1. 111,53"] rvvrnRv; sKRrK>; wttif a kkmainder 105 To show thai lim luuxaiudur iii>\ zero, let us write it in the Cauchy form : The first factor m(m-l) -(m-n) 1.2...n ^ approaches zero since it is the getieral term of a convergent series. The second factor (1 — 0) /(I -\- Ox) is less than unity; and, finally, the last factor (1 -f Ox)"'-^ is less than a fixed limit. For, if w - 1 > 0, we have (1 + Ar)"— * < 2"*-^ ; while if 7/t - 1< 0, (1 -I- ^)"*"'< (1 — Ixl)"-'. Hence for every value of x between — 1 and + 1 we have the development (16) (1+^)^ = 14-^x4- ^<7^^> x« + ... I rn(m-l).-.(m-n + l) ^^ ^ 1 .2---n We shall postpone the discussion of the case where x = ±1, In the same way we might establish the following formulae : 1.3.5 ••(2n-l) g«"^^ "*■ 2.4.6..2n 2»4-l x*x^ x'' , , . .. ar«-^» . arc tan x = x — — + — — — H 1-(— 1)" 3 5 7 • ^ ^ 2»4-l ' which we shall prove later by a simpler process, and which hold for all values of x between — 1 and 4- 1. Aside from these examples and a few others, the discussion of the remainder presents great difficulty on account of the increas- ing complication of the successive derivatives. It would therefore seem from this tirst examination as if the application of Taylor's series for the development of a function in an infinite series were of limited usefulness. Such an impression would, however, be utterly false ; for these developments, quite to the contrary, play a funda- mental rSle in modern Mathematical Analysis. In order to appre- ciate their importance it is necessar}' to take another point of view and to stuily the properties of power series for their own 1Q6 TAYLOR'S SERIES [III, § BO sake, irrespective cf their origin. We shall do this in several of the following chapters. Just now we will merely remark that the series /(o) + f /'(O) + i^/"(0) + • •• + r^/'"'(o) + • • • may very well be convergent without representing the function f{x) from which it was derived. The following example is due to Cauchy. Let f{x) = e -'^< . Then f'(x) = (2/x*)e-'^'^ ; and, in general, the nth derivative is of the form P -\ ^(n)(^)= e ^, X" where P is a polynomial. All these derivatives vanish for x = 0, for the quotient of e"^^"^ by any positive power of x approaches zero with x* Indeed, setting x = l/z, we may write Lri e ^ = e^' and it is well known that e*'/^"* increases indefinitely with z, no matter how large m may be. Again, let ^ (x) be a function to which the formula (8) applies : X" ♦(0=) = ♦(<)) + J *'(0) + • • ■ + j-g:^ *<">(0) + . . •. Setting F(x) = <>(x) + e'^"", we find f(0) = *(0), F'(0) = *'(0), •••, F<">(0) = ,^<"'(0), ••., and hence the development of F(x) by Maclaurin's series would coincide with the preceding. The sum of the series thus obtained represents an entirely different function from that from which the series was obtained. In gftieral, if two distinct functions f(x) and <^ (x), together with all their derivatives, are equal for a? = 0, it is evident that the *It U tacitly AMnmed that /(O) = 0, which is the only assignment which would nad»r/(x) oontlnuous at z = 0. But it should be noticed that no further assignment b mw mu r j for /'(a-), etc., at as = 0. For 11m /(g)~/(0) _ X /^W=a?=0 ^'r-''' =0> which <l«aiiM/'(jr) at as ■;= and makes /'(z) continuous at z = 0, etc. —Trans. Ill, §61] TAYLOR'S SEHIKS WITH A KKMAINDEB 107 Maclaurin series developments for the two functions cannot both be valid, for the coefficients of the two developments coincide. 51. Extension to functions of several variables. Let us consider, for detiniteness, a fuiictiou <i> =/(-c, y, ar) of the three independent vari- ables Xf y, z, and let us try to develop /(ar -f A, y -H k, z -\- 1) accord- ing to powers of h, ky /, grouping together the terms of the same degree. Cauchy reduced this problem to the preceding by the fol- lowing device. Let us give x, y, «, h, k, I definite values and let us set <^ (t) =f(x -f Af, y -f- kt, z -h U), where t is an auxiliary variable. The function <f>(t) depends on t alone ; if we apply to it Taylor's series with a remainder, we find (17) 4,(t) = *(0) + J .^'(0) + j^ *"(0) + • .» + l ^'"XO) + , o /^_.ix ^^"-^ *X^)' 1.2...n^ ^ ^ 1.2--(n-fl) where <^(0), <^'(^)> "» <^^"X^) *^® ^^® values of the function <f>{t) and its derivatives, for ^ = 0; and where <f>^'*'*'^\$t) is the value of the derivativiB of order n -f- 1 for the value Ot, where 6 lies between zero and one. But we may consider <^ (^) as a composite function of ty ifi(t) =/(m, V, w)y the auxiliary functions u = X i- ht, V = y -{- kt, w = z -{- It being linear functions of t. According to a previous remark, the expression for the differential of order m, rf*<^, is the same as if u, V, w were the independent variables. Hence we have the symbolic equation \cu ov cw J \cu cv cw I which may be written, after dividing by rff", in the form For f = 0, u, r, w reduce, respectively, to x, y, «, and the above equation in the same symbolism becomes *'-<». -(g-'^- If')"' IQg TAYLOR'S SERIES [III, §52 Similarly, where x, y, « are to be replaced, after the expression is developed, by X + Bht, y + 6kt, z + Bit, lespectiTely. If we now set ^ = 1 in (17), it becomes The remainder R^ may be written in the form " 1.2 where a?, y, « are to be replaced hy x -\- Oh^ y ■\- Ok, z + 61 after the expression is expanded.* This formula (18) is exactly analogous to the general formula (3). If for a given set of values of x, y, z, h, k, I the remainder R^ approaches zero when n increases indefinitely, we have a develop- ment of fix + A, y 4- A;, « H- Z) in a series each of whose terms is a homogeneous polynomial in A, k, I. But it is very difficult, in gen- eral, to see from the expression for R^ whether or not this remainder approaches zero. 52. From the formula (18) it is easy to draw certain conclusions analogous to those obtained from the general formula (3) in the case of a single independent variable. For instance, let z =f(x, y) be the equation of a surface S. If the function /(x^ y), together with all its partial derivatives up to a certain order n, is continuous in the neighborhood of a point (xq, yo), the formula (18) gives A*+»,y.+*)-A..,,.) + (ig + *g) Reatricting oumelyes, in the second member, to the first two terms, then to the first three, etc., we obtain the equation of a plane, then • It IsMMUMd here that all the derlvatlvea used exist and are continuous. — Trans. in. §52] TAYLOR'S SERIES WITH A REMAINDER 109 that of a paralx)loid, etc., which differ very little from the given sur- face near the point (xq, yo). The plane in question is precisely the tangent plane ; and the paraboloid is that one of the family « = ilx* + 2 Bxy -\-Cy* + 2Dx + 2Ei/-\-F which most nearly coincides with the given surface 5. The formula (18) is also used to determine the limiting value of a function which is given in indeterminate form. Let /(x, y) and <^ (x, y) be two functions which both vanish for x = a, y = by but which, together with their partial derivatives up to a certain order, are continuous near the point (a, b). Let us try to find the limit approached by the ratio f(^. y) when X and y approach a and 6, respectively. Supposing, first, that the four first derivatives df/da, df/dbj dif»/day c<^/db do not all vanish simultaneously, we may write •(K-)-(l'-') f(a -{.h,b-\-k) _ it,(a -{- h, b -\- k) /d<f> \ /cif> (5*-.)"(S-0 where e, e', c,, «[ approach zero with h and k. When the point (ar, y) approaches (a, b), h and k approach zero ; and we will sup- pose that the ratio k/h approaches a certain limit a, i.e. that the point {Xy y) describes a curve which has a tangent at the point (a, b). Dividing each of the terms of the preceding ratio by A, it appears that the fraction /(x, y)/<f>(Xf y) approaches the limit da db da db This limit depends, in general, upon <r, i.e. upon the manner in which X and y approach their limits a and i, respectively. In order that this limit should be independent of a it is necessary that the relation da db db da should hold ; and such is not the case in generaL IIQ TAYLOR'S SERIES [III, §63 If the four first derivatives a//aa, dfj^h, d^jda, d4>/dh vanish simultaneously, we should take the terms of the second order in the formula (18) and write where c, c', c", €„€,', c'/ are infinitesimals. Then, if a be given the same meaning as above, the limit of the left-hand side is seen to be which depends, in general, upon a. n. SINGULAR POINTS MAXIMA AND MINIMA 53. Singular points. Let (xqj yo) be the coordinates of a point Mq of a curve C whose equation is F(x, y) = 0. If the two first par- tial derivatives dF/dx, dF /dy do not vanish simultaneously at this point, we have seen (§ 22) that a single branch of the curve C passes through the point, and that the equation of the tangent at that point is where the symbol d^-^'^F/dxl dyl denotes the value of the derivative a^ ♦«F/3a?»' dj/f for a; = Xo, y = yo- If dF/dxo and dF/dy^ both van- ish, the point (^o. y©) is, in general, a singular point* Let us suppose that the three second derivatives do not all vanish simultaneously for x^X9,y = yo» aiid that these derivatives, together with the third deriTatiTes, are continuous near that point. Then the equation of the ounre may be written in the form. * That to, tlM ftppeanncA of the curve is, in general, peculiar at that point. For an aaalytJe dafloitloD of a ringular point, see $ 192. — Tkans. Ill, §wj SINGULAR POINTS MAXIMA AND MINIMA 111 ifd^F d*F d^F "I 1 VdP dp T" + r2li\ji ^' -''^ + Vy^!'- ^">J;;:jji;j;' where x and 3/ are to be replaced in the third derivatives by Xo-\-$(x — Xo) and yo-h 0(i/ — yo)y respectively. We may assume that the derivative d^F/dyl does not vanish; for, at any rate, we could always bring this about by a change of axes. Then, setting y — yo = t(^ — ^o) and dividing by (x — Xoy, the equation (19) becomes d^F d*F d^F (20) a;^ + 2.a^^ + ''ai^+(— »)^(--'^'') = «' where P{x — XQjt) is a function which remains finite when x approaches Xq. Now let ti and t^ be the two roots of the equation axj axo dy^ dy^ If these roots are real and unequal, i.e. if / d^F y d^Fd^F \dxQ dyj cxl dyi ' the equation (20) may be written in the form ^(^ - ty){t - U) ^{X-X,)P = 0. For X = Xq the above quadratic has two distinct roots < = ^1, t = t^. As X approaches 2*0 that equation has two roots which approach ti and fiy respectively. The proof of this is merely a repetition of the argument for the existence of implicit functions. Let us set t = ti -\- u, for example, and write down the equation connecting x and u : u(h _ ^, 4. u) -f (x - Xo) (2(x, tt) = 0, where Q(x, u) remains finite, while x approaches x© and u approaches zero. Let us suppose, for definiteness, that t^ — tt>0; and let M denote an upper limit of the absolute value of Q(x, t/), and m a lower limit of fi — ^j -f u, when x lies between x© — A and x© -f ^ 112 TAYLOR'S SERIES [III, §53 and 11 between — h and 4- A, where A is a positive number less than tx — tf. Now let c be a positive number less than h, and rj another positive Dumber which satisfies the two inequalities If :c be given such a value that |a; — xd is less than rj, the left-hand side of the above equation will have different signs if — c and then 4- c be substituted for u. Hence that equation has a root which approaches zero as x approaches Xq, and the equation (19) has a root of the form y = yo 4- (a; - Xo) (tt -f- or), where a approaches zero with x — Xq. It follows that there is one branch of the curve C which is tangent to the straight line y — yo = h(x- Xo) at the point (x©, y©)- In like manner it is easy to see that another branch of the curve passes through this same point tangent to the straight line y — t/o ~ ^t(^ — ^o)' The point Mq is called a double point; and the equation of the system of tangents at this point may be found by setting the terms of the second degree in (x — Xq), Q/ — y^) in (19) equal to zero. If \dx,dyj dxl dif. <o, the point (ajo, y©) is called an isolated double point. Inside a suffi- ciently small circle about the point Mq as center the first member F(ar, y) of the equation (19) does not vanish except at the point Mq itself. For, let us take 35 = a^o + p cos <^, y = ?/o + /» sin <^ as the cottrdinates of a point near M^. Then we find where L remains finite when p approaches zero. Let // be an upper limit of the absolute value of L when p is less than a certain posi- tive number r. For all values of <^ between and 27r the expression d^F , ^ d^F d^F in,§53] SINGULAR POINTS MAXIMA AND MINIMA 118 has the same sign, since its rcx)t8 are imaginary. Let m be a lower limit of its absolute value. Then it is clear that the coefficient of p* cannot vanish for any point inside a circle of radius p<m/H. Hence the equation F(x, y) = has no root other than /> = 0, i.e. X = a;©, y = 2/o> inside this circle. In case we have F /_?!£. V= ^ ?H the two tangents at the double point coincide, and there are, in gen- eral, two branches of the given curve tangent to the same line, thus forming a cusp. The exhaustive study of this case is somewhat intricate and will be left until later. Just now we will merely remark that the variety of cases which may arise is much greater than in the two cases which we have just discussed, as will be seen from the following examples. The curve y^ = x* has a cusp of the first kind at the origin, both branches of the curve being tangent to the axis of x and lying on different sides of this tangent, to the right of the y axis. The curve y^ — 2 x^y -\- X* — x^ =^ has a cusp of the second kindj both branches of the curve being tangent to the axis of x and lying on the same side of this tangent; for the equation may be written y = x^ dt x*f and the two values of y have the same sign when x is very small, but are not real unless x is positive. The curve x* -{- x^y^ - 6 x^y -\- y^ = has two branches tangent to the x axis at the origin, which do not possess any other peculiarity ; for, solving for y, the equation becomes 3x*±x^y/S-x* 2^ = iTV^ ' and neither of the two branches corresponding to the two signs before the radical has any singularity whatever at the origin. It may also happen that a curve is composed of two coincident branches. Such is the case for the curve represented by the equation F{x,y) = y^-2x*y + x*==0. When the point (x, y) passes across the curve the first member F(x, y) vanishes without changing sign. 114 TAYLOR'S SERIES [III, §54 Finally, the point (xo, yo) may be an isolated double point. Such is the case for the curve 1/ -\- x* + y* = 0, on which the origin is an isolated double point. 54. In like manner a point Mq of a surface 5, whose equation is F(Xy y, «) = 0, is, in general, a singular point of that surface if the three first partial derivatives vanish for the coordinates Xq, ijqj Zq of that point: |£ = o, 1^ = 0, 1^ = 0. dxQ dy^ czq The equation of the tangent plane found above (§ 22) then reduces to an identity ; and if the six second partial derivatives do not all vanish at the same point, the locus of the tangents to all curves on the surface S through the point Mq is, in general, a cone of the second order. For, let be the equations of a curve C on the surface S. Then the three functions f(t)f <f>(t), \l/(t) satisfy the equation F(x, y, z) = 0, and the first and second differentials satisfy the two relations dF dF dF (dp dF dF \"' dF dF dF For the point x = Xq, y = y^, z = Zq the first of these equations reduces to an identity, and the second becomes 3*F d^F d^F ^ ^yi ^4 ^ d*F d*F d^F "*■ ^ ?r^ dxdy + 2 r— r- dyd:i^2 ^-^ dx dz = 0. The equation of the locus of the tangents is given by eliminating dXf dy, dx between the latter equation and the equation of a tangent line dx dy dz ^ which leads to the equation of a cone T of the second degree : in, 5 54] SINGULAR POINTS MAXIMA AND MINIMA 115 (21) On the other hand, applying Taylor's series with a remainder and carrying the development to terms of the third order, the equa- tion of the surface becomes (22) 1 V»F 8F dF T" where ar, y, z in the terms of the third order are to be replaced by Xq-\- ${x — Xo)f yo + 6{ij — i/o), Zo + 6(z — Zq), respectively. The equation of the cone T may be obtained by setting the terms of the second degree in x — a*©, y — yoj z — ZqIu the equation (22) equal to zero. Let us then, first, suppose that the equation (21) represents a real non-degenerate cone. Let the surface .S and the cone T be cut by a plane P which passes through two distinct generators (i and G' of the cone. In order to find the equation of the section of the sur- face S by this plane, let us imagine a transformation of coordinates carried out which changes the plane /* into a plane parallel to the xy plane. It is then sufficient to substitute » = Zom the equation (22). It is evident that for this curve the point Mq is a double point with real tangents ; from what we have just seen, this section is composed of two branches tangent, respectively, to the two generators G, 6". The surface S near the point 3/o therefore resembles the two nappes of a cone of the second degree near its vertex. Hence the point i/© is called a conical point. When the equation (21) represents an imaginary non-degenerate cone, the point ^fo is an isolated singular point of the surface ^'. Inside a sufficiently small sphere about such a point there exists no set of solutions of the equation F(x, y, «) = other than x = Xq, y = yo> « = -0- For? let M be a point in space near 3/©, p the 116 TAYLOR'S SERIES [III, §55 distance AfA/o, and a, /8, y the direction cosines of the line MqM. Then if we substitute the function F(Xf y, z) becomes where L remains finite when p approaches zero. Since the equation (21) represents an imaginary cone, the expression cannot vanish when the point (a, ft y) describes the sphere a'^ + iS" + y' = 1. Let m be a lower limit of the absolute value of this polynomial, and let i/ be an upper limit of the absolute value of L near the point Mq. If a sphere of radius m/H he drawn about Mq as center, it is evident that the coefficient of p^ in the expression for F(x, y, z) oannot vanish inside this sphere. Hence the equation F(x, y, «) = has no root except p = 0. When the equation (21) represents two distinct real planes, two nappes of the given surface pass through the point Mq, each of which is tangent to one of the planes. Certain surfaces have a line of double points, at each of which the tangent cone degenerates into two planes. This line is a double curve on the surface along which two distinct nappes cross each other. For example, the circle whose equations are « = 0, x* + y* = 1 is a double line on the surface whoee equation is «* 4- 2z^(x* + y2) _ (a;2 + y2 _ ly ^ q. When the equation (21) represents a system of two conjugate im ag inar y planes or a double real plane, a special investigation is neoetsary in each particular case to determine the form of the sur- face near the point 3/©. The above discussion will be renewed in the paragraphs on extrema. 66. Extrema of functions of a single variable. Let the function f(x) be oontinuuus in the interval (a, b), and let c be a point of that I1I,$MJ SINGULAR POINTS MAXIMA AND MINIMA 117 interval. The function f{x) is said to have an eoUremum (Le. a maximum or a minimum) for 05 = c if a positive number iy can be found such that the difference f{c -\- h) —f{c), which vanishes for A = 0, has the same sign for all other values of h between — 17 and -f -q. If this difference is positive, the function f(x) has a smaller value for x = c than for any value of x near c; it is said to have a minimum at that point. On the contrary, if the differ- ence /(c -f h) —f{c) is negative, the function is said to have a maximum. If the function f{x) possesses a derivative for x = c, that deriva- tive must vanish. For the two quotients nc + h)-f{c) f(c-.h)-f(c) h -h each of which approaches the limit /'(c) when h approaches zero, have different signs ; hence their common limit /'(c) must be zero. Con- versely, let c be a root of the equation /'(x) = which lies between a and b, and let us suppose, for the sake of generality, that the first derivative which does not vanish for x = c is that of order n, and that this derivative is continuous when x = c. Then Taylor's series with a remainder, if we stop with 71 terms, gives /(<= + A) -/(«) = r:|!^/*"H<' + «A), which may be written in the form • /(« + A) -/(c) = j-|^ [/<•>(.) + .], where c approaches zero with A. Let ly be a positive number such that \f^*\c) I is greater than c when x lies between c — 1; and c -f 17. For such values of x, /^"^c) -}- c has the same sign as f^*^(c), and consequently /(c -f h) —f(c) has the same sign as A"/<"^(c). If n is odd, it is clear that this difference changes sign with A, and there is neither a maximum nor a minimum at x = c. If n is even, /(c -}- A)— /(c) has the same sign as/^">(c), whether A be positive or negative ; hence the function is a maximum if /^"^(c) is negative, and a minimum if f^*\c) is positive. It follows that the necessary and sufficient condition that the function /(x) should have a maximum or a minimum f or aj = c is that the first derivative which does not vanish f or x == c sliould be of even order. 118 TAYLOR'S SERIES [III, §56 Geometrically, the preceding conditions mean that the tangent to the curve y =/(a?) at the point A whose abscissa is c must be par- allel to the axis of ar, and moreover that the point A must not be a point of inflection. NUm, When the hypotheses which we have made are not satisfied the function f{x) may have a maximum or a minimum, although the derivative /'(x) does not vanish. If, for instance, the derivative is infinite for a; = c, the function will have a maximum or a mini- mum if the derivative changes sign. Thus the function y = x^ is at a minimum for a; = 0, and the corresponding curve has a cusp at the origin, the tangent being the y axis. When, as in the statement of the problem, the variable x is restricted to values which lie between two limits a and ^, it may happen that the function has its absolute maxima and minima pre- cisely at these limiting points, although the derivative /'(a:r) does not vanish there. Suppose, for instance, that we wished to find the shortest distance from a point P whose coordinates are (a, 0) to a circle C whose equation is x^ -\- y'^ — R^ =i 0. Choosing for our independent variable the abscissa of a point M of the circle C, we find (? = pF^ = (a; - a)2 + y2 ^ x2 4- y2 _ 2 «a; H- a" , or, making use of the equation of the circle, d^ = R''^-a'-2ax. The general rule would lead us to try to find the roots of the derived equation 2 a = 0, which is absurd. But the paradox is explained if we observe that by the very nature of the problem the variable x must lie between — R and -f 72. If a is positive, d?^ has a minimum iox x = R and a maximum for a; = — R. 56. Eztrema of functions of two variables. Let /(x, y) be a con- tinuous function of x and y when the point A/, whose coordinates are x and y, lies inside a region O bounded by a contour C. The function f{x, y) is said to have an extremum at the point M^ (a;©, yo) of the region O if a positive number y^ can be found such that the diffeMnoe ^ =/(a;o -I- h, yo + k) -f(xoy yo), which Tanishes for h = k = 0, keeps the same sign for all other sets of values of the increments A and k which are each less than ,; in lil,§5«ij SINGLLAK I'OINTS MAXIMA AND MINIMA 119 absolute value. Considering y for the moment as constant and equal to yo> « becomes a function of tlie single variable x ; and, by the above, the difference /(xo 4- A, yo) -/(xo, yo) cannot keep the same sign for small values of h unless the deriva- tive df/dx vanishes at the point M^. Likewise, the derivative df/dy must vanish at Mq ; and it is apparent tliat the only possible sets of values of x and y which can render the function /(ar, y) an extre- mum are to be found among the solutions of the two simultaneous equations ox cy Let a; = Xq, y = yo ^ a- set of solutions of these two equations. We shall suppose that the second partial derivatives of /(x, y) do not all vanish simultaneously at the point 3/o whose coordinates are (xq, y^), and that they, together with the third derivatives, are all continuous near 3/y. Then we have, from Taylor's expansion, ^ =/(^o -\-h,y^ + k) -/(j-o, yo) (23) \ 1.2\ dxi «x^cy„ cyi/ We can foresee that the expression cxl ox^cy^ cy^ will, in general, dominate the whole discussion. In order that there be an extremum at Mq it is necessary and sufficient that the ditference A should have the same sign when the point (xq -f a, yo + k) lies anywhere inside a sufficiently small square drawn about the point Mq as center, except at the center, where A = 0. Hence A must also have the same sign when the point (xq -f- a, yo + k) lies anywhere inside a sufficiently small circle whose center is 3fo; for such a square may always be replaced by its inscribed circle, and conversely. Then let C be a circle of radius /' drawn about the point Mq as center. All the points inside this circle are given by h = pcos<f>, k = psui<f>. 120 TAYLOR'S SERIES [III, §56 if-here ^ is to vary from to 2 tt, and p from — r to + r. We might, indeed, restrict p to positive values, but it is better in what follows not to introduce this restriction. Making this substitution, the expression for A becomes A = ^ (^ C08*«^ -f 2 J5 sin <^ cos <^ + C* sin^c^) + ^ ^ where and where Z, is a function whose extended expression it would be useless to write out, but which remains finite near the point (iCo, yo)« It now becomes necessary to distinguish several cases according to thesignof B«-^C. Firtt case. Let B^ — AC >0. Then the equation A cos^<l> + 2 B sin <^ cos <^ + C sin^<^ = has two real roots in tan <f>, and the first member is the difference of two squares. Hence we may write ^ = 2 ['^ (" cos <^ + ^ sin </>)2 - I3(a' cos <f> -\- b' sin <^)2] +^L, where a: > 0, ^ > 0, ab'- ba' ^ 0. If ^ be given a value which satisfies the equation a cos <^ + J sin <^ = 0, A will be negative for sufficiently small values of p ; while, if <^ be such that a'cos<^ -1- 6'sin<^ = 0, A will be positive for infinitesimal yalues of p. Hence no number r can be found such that the differ- ence A has the same sign for any value of <^ when p is less than r. It follows that the function f(x, y) has neither a maximum nor a minimum for x = Xoj y = y^. Seeandease. LetB*- AC<0. The expression A co8*<t> -f 2 fi cos<^ 8in<^ + C 8in*<^ oannot vanish for any value of <!>. Let m be a lower limit of its absolute value, and, moreover, let // be an upper limit of the abso- lute value of tlie funcrtion L in a circle of radius K about (aro, yo) aa ^V^^O, III, $57] SiMiLLAU I'UlNTS M.VAIMA AM) M1M31.\ 121 center. Finally, let r denote a positive number less than R and less than 3m/ II. Then inside a circle of radius r the difference A will have the same sign a^ the coefiicient of p'j i.e. the same sign as A or ( '. Hence the function /(x, y) has either a mail mum or a mini- iiiuin for X = Xq, y = yo. To recapitulate, if at the point (a;©, yo) we have \^XodyJ dxi dyi ' there is neither a maximum nor a minimum. But if /jyy Y^XodyJ dxl dyi there is either a maximum or a minimum, depending on the sign of the two derivatives ^//^xj* ^f/^^- There is a maximum if these derivatives are negative, a minimum if they are positive. 57. The ambiguous case. The case where B^ — AC = is not cov- ered by the preceding discussion. The geometrical interpretation shows why there should be difficulty in this case. Let S be the surface represented by the e(iuation ;;; =/(a", y). If the function f(x, y) has a maximum or a minimum at the point (x^, yo), near which the function and its derivatives are continuous, we must have which shows that the tangent plane to the surface S at the point iUo, whose coordinates are (xq, yo, Zq)^ must be parallel to the xy plane. In order that there should be a maximum or a minimum it is also necessary that the surface 5, near the point Mq, should lie entirely on one side of the tangent plane ; hence we are led to study the behavior of a surface with respect to its tangent plane near the point of tangency. Let us suppose that the point of tangency has been moved to the origin and that the tangent plane is the xy plane. Then the equa- tion of the surface is of the form (24) z = ax^-\- 2bxy + cy« + ax* -f Spx^y + 3 yxy* + 8/, where a, b, r are constants, and where a, p, y, S are functions of x and y which remain finite when x and y approach zero. This equa- tion is essentially the same as equation (19), where Xo and y© have been replaced by zeros, and h and k by x and y, respectively. 122 TAYLOR'S SERIES [in,§W In order to see whether or not the surface S lies entirely on one side of the xi/ plane near the origin, it is sufficient to study the section of the surface by that plane. This section is given by the equation (25) ax^ -\-2bxi/ + cy^ + ax*+-- = 0j hence it has a double point at the origin of coordinates. If b^ — ac is negative, the origin is an isolated double point (§ 53), and the equation (25) has no solution except x = y = 0, when the point (x, y) lies inside a circle C of sufficiently small radius r drawn about the origin as center. The left-hand side of the equation (25) keeps the same sign as long as the point (a;, y) remains inside this circle, and all the points of the surface S which project into the interior of the circle C are on the same side of the xy plane except the origin itself. In this case there is an extremum, and the por- tion of the surface S near the origin resembles a portion of a sphere or an ellipsoid. If b^ — ac> Of the intersection of the surface S by its tangent plane has two distinct branches Cu C^ which pass through the origin, and the tangents to these two branches are given by the equation ax^ + 2 bxy -f- cy^ = 0. Let the point (x, y) be allowed to move about in the neighborhood of tlie origin. As it crosses either of the two branches Ci, Cg, the left-hand side of the equation (25) vanishes and changes sign. Hence, assigning to each region of the plane in the neighborhood of the origin the sign of the left-hand side of the equation (25), we find a configuration similar to Fig. 7. Among the points of the surface which project into points inside a circle about the origin in the xy plane there are always some which lie below and some which lie above the xy plane, no matter how small the circle be taken. The general aspect of the sur- face at this point with respect to its tan- gent plane resembles that of an unparted hyperboloid or an hyperbolic paraboloid. The function /(«, y) has neither a maxi- mum nor a minimum at the origin. The oase where &• — a<; = is the case in which the curve of interseotion of the surface by its tangent plane has a cusp at the origin. We will postpone the detailed discussion of this case. If the Fio.7 111,558] SINGULAR POINTS MAXIMA AND MINIMA 128 intersection is composed of two distinct branches through the origin, there can be no extremiun, for the surface again cuts the tangent plane. If the origin is an isolated double point, the function /(x, y) has an extremum for x = y = 0. It may also happen that the inter- section of the surface with its tangent plane is composed of two coincident branches. For example, the surface z = y* — 2 x*y -f x* is tangent to the plane « = all along the parabola y = x*. The function y* — 2 x^y 4- x* is zero at every point on this parabola, but is positive for all points near the origin which are not on the parabola. 88. In order to see which of these cases holds in a given example it is neces- Kury to take into account the derivatives of the third and fourth orders, and some- times derivatives of still higher order. The following discussion, which is usually sufficient in practice, is applicable only in the most general cases. When 6* — oc = the equation of the surface may be written in the following form by using Taylor's development to terms of the fourth order: (26) z =/(x, y) = A (xsinw - ycosu,)^ + i>s{x, y) + -\ (»ii + v!r)»x- z-i \ cz cy / gg Let us suppose, for definiteness, that A is positive. In order that the surface S should lie entirely on one side of the xy plane near the origin, it is necessary that all the curves of intersection of the surface by planes through the z axis should lie on the same side of the xy plane near the origin. But if the surface be cut by the secant plane y = ztan0, the equation of the curve of intersection is found by making the substitution x = p cos 0, y = psm<f> in the equation (26), the new axes being the old z axis and the trace of the secant plane on the xy plane. Performing this operation, we find z = Afl^ (cos sin w — cos « sin 0)^ + Kp* + L^, where K is independent of p. If tan « ^ tan 0, z is positive for sufficiently small values of p ; hence all the corresponding sections lie above the xy plane near the origin. Let us now cut the surface by the plane y = x tan m. If the corresponding value of X is not zero, the development of e is of the form z = pfi{K + t) and changes sign with p. Hence the section of the surface by this plane has a point of inflection at the origin and crosses the xy plane. It follows that the function /(x, y) has neither a maximum nor a minimum at the origin. Such is the case when the section of the surface by its tangent plane has a cusp of the first kind, for instance, for the surface « = y« - x«. 124 TAYLOR'S SERIES [III, §68 If JT = for the latter lubstitution, we would carry the development out to tann* of the fourth order, and we would obtain an expression of the form where ITi ii a constant which may be readily calculated from the derivatives of the fourth order. We shall suppose that Ki is not zero. For infinitesimal val- ues of pt t has the same sign as iTi ; if Ki is negative, the section in question lies beneath the zy plane near the origin, and again there is neither a maximum nor a minimum. Such is the case, for example, for the surface z = y^ — x*, whose intenection with the xy plane consists of the two parabolas y = ±x'^. Hence, ntili^a K = and iri>0 at the same time, it is evidently useless to carry the inresUgation farther, for we may conclude at once that the surface crosses its tangent plane near the origin. But if £* = and Ki>Q at the same time, all the sections made by planes throng the z axis lie above the xy plane near the origin. But that does not ■how conclusively that the surface does not cross its tangent plane, as is seen by considering the particular surface z = {y-x'^){y-2x% wliich cuts its tangent plane in two parabolas, one of which lies inside the other. In order tliat the surface should not cross its tangent plane it is also necessary that the section of the surface made by any cylinder whatever which passes through the z axis should lie wholly above the xy plane. Let y = <f>{x) be the equation of the trace of this cylinder upon the xy plane, where {x) vanishes for X = 0. The function F{x) =/[x, <f>{x)] must be at a minimum for a; = 0, what- ever be the function <f> (x). In order to simplify the calculation we will suppose that the axes have been so chosen that the equation of the surface is of the form where A is positive. With this system of axes we have ««o ' ^0 ' dxl ' dxodyo ' dyl ' at the origin. The derivatives of the function F{x) are given by the formulae Ul,$fiO] SINGULAR POINTS MAXUiA AND MINIMA 125 from which, for x = y = 0, we obtain F'(0) = 0, F"(0) = ^[^'(0)]«. If 0'(O) does not yanlsb, the function F(x) has a minimum^ a« is aljo apparent from the previous discussion. But if 4>'(0) = 0, we find the formula ^(0) = 0, F"(0) = 0, F'"(0) = ^, Hence, in order that F(x) be at a minimum, it is necessary that d^f/ta^ vanish and that the following quadratic form in 0"(O), be positive for all values of 4>^'{0). It is easy to show that these conditions are not satisfied for the above function « = y* — 3x'*y + 2x*, but that they are satisfied for the function « = y* + a:*. It is evident, in fact, that the latter surface lies entirely above the xy plane. We shall not attempt to carry the discussion farther, for it requires extremely nice reasoning to render it absolutely rigorous. The reader who wishes to exam- ine the subject in greater detail is referred to an important memoir by Ludwig Schefler, in Vol. XXXV of the Mathematische Annalen. 59. Functions of three variables. Let u = f(x, t/, z) he 2l continuoiia function of the three variables a*, y, z. Then, as before, this func- tion is said to have an extremum (maximum or minimum) for a set of values Xq, i/o, Zq if a positive number ri can be found so small that the difference ■ ^ =/(a'o + h, 2/0 + A', ^0 4- -/(a-o, yo, «o), which vanishes for A = A; = Z = 0, has the same sign for all other sets of values of A, k^ /, each of which is less in absolute value than -q. If only one of the variables a*, y, z is given an increment, while the other two are regarded as constants, we find, as above, that tt cannot be at an extremum unless the equations are all satisfied, provided, of course, that these derivatives are con- tinuous near the point (jr^, y^, «„). Let us now suppose that x^, yo» *• are a set of solutions of these equations, and let Mq be the point whose coordinates are j-q, %, Zq. There will be an extremum if a sphere can be drawn about M^ so small that /(x, y, x) — f{x^y y©, «J 126 TAYLOR'S SERIES [III, §59 has the same sign for all points (x, y, z) except Mo inside the sphere. Let the coordinates of a neighboring point be represented by the equations X = aro + par, y = y© + pA « = «o + py, where a, A y satisfy the relation a« + /3^ + / = 1 ; and let us replace a; _ Xo, y - yo, « - «o in Taylor's expansion of /(a;, y, «) by pa, p/3, py, respectively. This gives the foUowing expression for A : A = p*[<^(a, Ay) + p^], where <^(a, ft y) denotes a quadratic form in a, ft y whose coeffi- cients are the second derivatives of f(x, y, ^), and where Z is a function which remains finite near the point M^. The quadratic form may be expressed as the sum of the squares of three distinct linear functions of a, ft y, say P, P', P", multiplied by certain con- stant factors a, a', a", except in the particular case when the dis- criminant of the form is zero. Hence we may write, in general. <f>(cx, ft y) = aP^ + a'P'^ + a'^P"' where a, a', a" are all different from zero. If the coefficients a, a', a" have the same sign, the absolute value of the quadratic form </> will remain greater than a certain lower limit when the point a, ft y describes the sphere a' + i8* + y' = 1, and accordingly A has the same sign as a, a', a" when p is less than a certain number. Hence the function f(x, y, z) has an extremum. If the three coefficients a, a\ a" do not all have the same sign, there will be neither a maximum nor a minimum. Suppose, for example, that a > 0, a' < 0, and let us take values of a, ft y which satisfy the equations P' = 0, P" = 0. These values cannot cause P to vanish, and A will be positive for small values of p. But if, on the other hand, values be taken for a, ft y which satisfy the equa- tions P = 0, P" = 0, A will be negative for small values of p. The method is the same for any number of independent variables : the discussion of a certain quadratic form always plays the prin- cipal rale. In the case of a function u =f(x, y, z) of only three independent variables it may be noticed that the discussion is equivalent to the discussion of the nature of a surface near a singu- lar point. For consider a surface 2 whose equation is F(«, y, «) =/(x, y, «) -/(a^o, yo, «o) = ; UI,J60J SINGULAR POINTS MAXIMA AND MINIMA 127 this surface evidently passes through the point i/« whose co6rdi* nates are (Xo, yo> «o)> and if the function /(x, y, z) has an extremum there, the point ^f^, is a singular point of 2. Hence, if the cone of tangents at Mo is imaginary, it is clear that F(jr, y, z) will keep the same sign inside a sufficiently small sphere about J/q as center, and /{^y !/} «) will surely have a maximum or a minimum. But if the cone of tangents is real, or is composed of two real distinct planes, several nappes of the surface pass through A/o, and F(x, y, z) changes sign as the point (sc, y, z) crosses one of these nappes. 60. Distance from a point to a surface. r..et us try to find the maximum and the ininimuin values of tlie diKUiiice from a Hzed point (a, 6, c) to a surface S whoM equation is F(x, y, z) = 0. The square of this distance, u = da = (z - a)2 + (y - 6)« + (2 - c)«, is a function of two indei)endent variables only, — z and y, for example, if « be considered as a function of x and y defined by the equation F = 0. In order that u be at an extremum for a point (z, y, z) of the surface, we must have, for the coordinates of that point, 2 dy cy We find, in addition, from the equation F = 0, the relations dx dz dx~ ^ dy dz dy~ ^ whence the preceding equations take the form z — a y — b _ z — c dF ~ dJF d_F ' dz dy dz This shows that the normal to the surface iS at the point (x, y, z) passes through tlie point (a, 6, c). Hence, omitting the singular points of the surface 5, the points sought for are the feet of normals let fall from the point (a, 6, c) upon the surface S. In order to see whether such a point actually corresponds to a maxi- mum or to a minimum, let us take the point as origin and the tangent plane as the xy plane, so that the given point shall lie upon the axis of z. Then the func- tion to be studied has the form u = z« + y2 + (z - c)«, where z is a function of z and y which, together with both its first derivatives, vanishes for x = y = 0. Denoting the second partial derivatives of z by r, s, t, we have, at the origin, 128 TAYLOR'S SERIES [ni,§61 and it only remains to study the polynomial A(c) = c«ja - (1 - cr)(l - ct) = c^8^ -rt) + {r-ht)c - 1. The root* of the equation A(c) = are always real by virtue of the identity /|. 4. t)« ^. 4 («« - rf ) = 4 «« + (r - t)^. There are now several cases which must be distinguished according to the sign of a^ - rt. FMt case. Let »« - rf < 0. The two roots Ci and Ca of the equation A (c) = have the same sign, and we may write A (c) = (s^ - rt) (c - ci) (c - C2). Let us now mark the two points Ai and A^ of the z axis whose coordinates are Ci and Cj. Theee two points lie on the same side of the origin ; and if we suppose, as is always allowable, that r and t are positive, they lie on the positive part of the f axis. If the given point A (0, 0, c) lies outside the segment A1A2, A(c) is negative, and the distance OA is a maximum or a minimum. In order to see which of the two it is we must consider the sign of 1 - cr. This coefficient does not vanish except when c = 1 /r ; and this value of c lies between Ci and Cj, since A (1 /r) = 8'^/r^. But, f or c = 0, 1 — cr is positive ; hence 1 — cr is posi- tive, and the distance OA is a minimum if the point A and the origin lie on the same side of the segment A1A2. On the other hand, the distance OA in a maximum if the point A and the origin lie on different sides of that segment. When the point A lies between Ai and A2 the distance is neither a minimum nor a maximum. The case where A lies at one of the points Ai^ A^ is left in doubt. Second case. Let s* — ri > 0. One of the two roots Ci and C2 of A (c) = is positive and the other is negative, and the origin lies between the two points Ai and At. If the point A does not lie between Ai and A2, A(c) is positive and there is neither a maximum nor a minimum. If A lies between Ai and A%^ A(c) is negative, 1 — cr is positive, and hence the distance OA is a minimum. Third case. Let a^ -rt = 0. Then A (c) = {r + t) {c - Ci), and it is easily seen, as above, that the distance OA is a minimum if the point A and the origin lie on the same side of the point ^1, whose coordinates are (0, 0, Ci), and that there is neither a maximum nor a minimum if the point Ai lies between the point A and the orighi. The points Ai and A2 are of fundamental importance in the study of curva- tnre; they are the principal centers of curvature of the surface S at the point O. 61. MiTimfl and minima of implicit functions. We often need to find the maxima and minima of a function of several variables which are connected by one or more relations. Let us consider, for example, a function a> =/(«, y, z, it) of the four variables Xy y, «, w, which themselves satisfy the two equations /i (x, y, «, u) = 0, /,(x, y, «, u) = 0. For definiteness, let us think of x and y as the independent vari- ablei, and of z and u as functions of x and y defined by these equa- tion!. Then the necessary conditions that <u have an extremum are UI,i61] SINGULAB POINTS MAXIMA AND MIKIMA 129 dx dz dx du dx ' dy dz dy du dy * and the partial derivatives dz/dxj du/dx, dz/dy, du/dy are given by the relations dx dz dx du dx * dx dz dx du dx * dy dz dy du dy * dy dz dy du dy The elimination of dz/dx, du/dx, dz/dy^ du/dy leads to the new equations of condition '^^ ^ /)(x, z, u) ' Z>(y, z, u) ^' which, together with the relations /j = 0, /j = 0, determine the val- ues of a*, y, ;?, w, which may correspond to extrema. But the equa- tions (27) express the condition that we can find values of X and /i which satisfy the equations (28) dx dx dx ' dy dy dy * ^« ^« "^ <7« du du '^ c?u hence the two equations (27) may be replaced by the four equations (28), where A and /i are unknown auxiliary functions. The proof of the general theorem is self-evident, and we may state the following practical rule : Given a function /(a-i, a-j, . • • , X,) of n variables, connected by h distinct relations <^i = 0, <^,=:0 , <^A = 0; in order to find the values of x^, x^, ••, x^ xrhich may render this function an extremum xce must equate to zero the partial derivatives of the auxilia ry function f -\- K<^\ -r • • -\- K4>k} regarding Ai, A3, • • , X^ <w constants. 180 TAYLOR'S SERIES [III, §62 M. Aa«thtr example. We shall now take up another example, where the mini- mam is not necessarily given by equating the partial derivatives to zero. Given a triangle ABC ; let us try to find a point P of the plane for which the sum PA + PB + PC of the distances from P to the vertices of the triangle is a minimnm. Let (ci, 61), (oa, 6a), (08, bs) be respectively the coordinates of the ▼ertices A,B,C referred to a system of rectangular coordinates. Then the func- tion whose minimum is sought is (29) « = V(x- ai)a -\-{y- 61)2 +V(x- 03)2 + (y-ft2)^+V(x - as)^+{y -bs)^, where each of the three radicals is to be taken with the positive sign. This equa- tion (29) represents a surface S which is evidently entirely above the xy plane, and the whole question reduces to that of finding the point on this surface which is nearest the xy plane. From the relation (29) we find 9t : X-Oi -ai)* + (y- y-bi -ai)^ + (y- X - y- -aa)2 -Oa + {y- -62 + (y- -62)2 -1 X - as ax — = 4- -a8)2 + (y- y -bi -hr «F V(x- - a8)2 + {y - -w and it is evident that these derivatives are continuous, except in the neighbor- hood of the points A, B, C, where they become indeterminate. The surface 5, therefore, has three singular points which project into the vertices of the given triangle. The minimum of 2 is given by a point on the surface where the tan- gent plane is parallel to the xy plane, or else by one of these singular points. In order to solve the equations dz/dx = 0, dz/dy = 0, let us write them in the form X — a\ X — Oj X — az V(x - 01)2 + (y - 6i)2 V(x - aa)2 + (y - 62)^ V(x - a,)2+ (y - 63)2' y-bi y- 62 y -bz V'(»-ai)» + (y-6i)2 V(x - 03)2 + (y - 63)2 V(x - 03)2 + (y - 63)2* Then squaring and adding, we find the condition 1 \ 2 _(^«i)(^ - 02) 4- (y - 61) (y - &2)__ ^ ^ V(x - ai)2 + (y - 6i)2 V(x - 02)2 + (y - 62)^ The geometrical interpretation of this result is easy : denoting by a and p the eosines of the angles which the direction PA makes with the axes of x and y, respectively, and by a' and ^ the cosines of the angles which PB makes with the axes, we may write this last condition in the form H-2(aa' + /3/S0=O, or, denoting the angle APB by w, 2 cos w + 1 = 0. Bmm the condition in question expresses that the segment AB subtends an •agte of 120* at the point /*. For the same reason each of the angles BPC and OPA must be 120*».* It is clear that the point P must lie inside the triangle • The reader is urged to draw the figure. III,i63] SINGULAR POINTS MAXIMA AND MINIMA 131 ABC, and that there Lb no point which poMOMon the required property if any angle of the triangle ABC is equal to or greater than 120^. In case none of the angles is as great as 120°, the point P is uniquely determined by an easy ood« struction, as the intersection of two circles. In this case the minimum \m gbeo by the point P or by one of the vertices of the triangle. But it is easy to show that the sum PA + PB + PC is less than the sum of two of the sides of the tri- angle. For, since the angles APB and APC are each 120°, we find, from the two triangles PAC and PBA, the formulse AB- Va^ + b^ + od, ilC= Va«Tl«Tac, where PA = a, PB = 6, PC = c. But it is evident that 2 2 and hence AB-\- AOa-^b + c. The point P therefore actually corresponds to a minimum. When one of the angles of the triangle ABC is efjual to or greater than 120" there exists no point at which each of the sides of the triangle ABC subtends an angle of 120°, and hence the surface S has no tangent plane which is parallel to the zy plane. In this case the minimum must be given by one of the vertices of the triangle, and it is evident, in fact, that this is the vertex of the obtuse angle. It is easy to verify this fact geometrically. 68. D'Alembert's theorem. Let F(x, y) be a polynomial in the two variables X and y arranged into homogeneous groups of ascending order ^(x, y) = H + <Pp{x, y) 4- 0p + i(a:, y) + • • • + ^«(x, y), where fl" is a constant. If the equation <t>p (x, y) — 0, considered as an equation in y/x, has a simple root, the function F(x, y) cannot have a maximum or a mini- mum for X = J/ = 0. For it results from the discussion alx)ve that there exist sec- tions of the surface z + H = F(x, y) made by planes through the z axis, some of which lie above the xy plane and others below it near the origin. From this remark a demonstration of d'Alembert's theorem may be deduced. For, let/(z) be an integral polynomial of degree m, /(z) = ^0 -I- Axz + Atz* + • • . + An,z^, where the coefficients are entirely arbitrary. In order to separate the real and imaginary parts let us write this in the form /(x -f- xy) = oo + t6o + (ai + i6i) (X + ty) + • • • + (cu + «>»,) (z + <y)", where oo, bo, ai, &i, - • , ou, hm are real. We have then f{z) = P-\-iq, where P and Q have the following meanings : P = Oo + aix - feiv + • • • , Q = bo + 6ix + ail/ + • • . ; and hence, finally, |/(«)|=Vp«+<?. X32 TAYLOR'S SERIES [HI, §63 We will firtt show that I/(z)|, or, what amounts to the same thing, that P« + Q", cannot be at a minimum for x = y = except when a© = 60 = 0. For this purpose we shall introduce polar coordinates p and 0, and we shall suppose, for the sake of generality, that the first coefficient after Aq which does not Ttnish is Ap. Then we may write the equations P = Oo + (0,. cosp0 - 6p sinp0)p'' + • • • , Q = 60 + (ftp CO8P0 -}- Op sin p0)pP + ••• , pt+Qa = a; + 6; + 2pP [(ooOp + 60M cosjx^ + {bottp - aobp) ain p<f>] + • • • , where the terms not written down are of degree higher than p with respect to p. But the equation (ooOp + bobp) co8p<f> + (bottp - Oobp) 8mp<f> = gives tanp0 = X", which determines p straight lines which are separated by angles each equal to 2 it /p. It is therefore impossible by the above remark that P« + Q8 should have a minimum for x = y = unless the quantities OoOp + bobpy boOp — Oobp both vanish. But, since aj + 6J is not zero, this would require that ao = 60 = ; that is, that the real and the imaginary parts of f{z) should both vanish at the origin. If \/{z) I has a minimum for x = or, y = ^, the discussion may be reduced to the preceding by setting 2 = a + i/3 + 2'. It follows that \f{z)\ cannot be at a minimum unless P and Q vanish separately f or x = or, y = /3. The absolute value of f{z) must pass through a minimum for at least one value of z, for it increases indefinitely as the absolute value of z increases indefi- nitely. In fact, we have where the terms omitted are of degree less than 2 m in p. This equation may be written In the form Vp^-^(^ = p'n(y/a^ + bl + e), where t approaches zero as p increases indefinitel y. Henc e a circle may be drawn whose radius iJ is so large that the value of Vp* + Q2 jg greater at every point of the circumference than it is at the origin, for example. It follows that there is at least one point X = a, y = /3 this circle for which vP« -f Q3 is at a minimum. By the above it fol- that the point x = a, y = /3 is a point of intersection of the two curves P = 0, Q = 0, which amounts to saying that 2 = a + /Siis a root of the equation /(s)=0. In this example, as In the preceding, we have assumed that a function of the two variables x and y which is continuous in the interior of a limited region actually a ss nme s a minimum value inside or on the boundary of that region. This It a iUtement which will be readily granted, and, moreover, it will be rigorously demonstrated a little later (Chapter VI). Ill, Exs] EXERCISES 133 EXERCISES 1. Show that the number tf, which occurs In Lagrange*! form of the re- mainder, approaches the limit l/(n + 2) ajs A approaches zero, provided that /('• + «;(a) ia not zero. 2. Let F{x) be a determinant of order n, all of whoee element! are funetiona of X. Show that the derivative F\x) is the sum of the n determinants obtained by replacing, successively, all of the elements of a single line by their deriva- tives. State the corresponding theorem for derivatives of higher order. 3. Find the maximum and the minimum values of the distance from a fixed point to a plane or a skew curve ; between two variable points on two curves ; between two variable points on two surfaces. 4. The points of a surface S for which the sum of the squares of the dis- tances from n fixed points is an extremum are the feet of the normals let fall upon the surface from the center of mean distances of the given n fixed points. 5. Of all the quadrilaterals which can be formed from four given sides, Uiat which is inscriptible in a circle has the greatest area. State the analogous theorem for polygons of n sides. 6. Find the maximum volume of a rectangular parallelepiped inscribed in an ellipsoid. 7. Find the axes of a central quadric from the consideration that the vertices are the points from which the distance to the center is an extremum. 8. Solve the analogous problem for the axes of a central section of an ellipaoid. 9. Find the ellipse of minimum area which passes through the three vertices of a given triangle, and the ellipsoid of minimum volume which passes through the four vertices of a given tetrahedron. 10. Find the point from which the sum of the distances to two given straight lines and the distance to a given point is a minimum. [Joseph BaaTRAKD.] 11. Prov6 the following formulas : log(z + 2) = 2 log(x + 1) - 2 log (X - 1) -f log(x - 2) [BoROA*8 Series.] log(i + 6) = log(x + 4) + log(x + 3) - 2 logx + log(x - 3) + log(x - 4) - log(x - 6) af ^^ iV ^^ V, 1 L«*-26*« + 72 8Vx»-«6x« + 72/ J [Habo^s Seriea.] CHAPTER IV DEFINITE INTEGRALS I. SPECIAL METHODS OF QUADRATURE 64- Quadrature of the parabola. The determination of the area bounded by a plane curve is a problem which has always engaged the genius of geometricians. Among the examples which have come down to us from the ancients one of the most celebrated is Archimedes' quadrature of the parabola. We shall proceed to indicate his method. Let us try to find the area bounded by the arc A CB of a parabola and the chord AB. Draw the diameter CD, joining the middle point D of AB tx> the point C, where the tangent is parallel to AB. Connect AC and BC, and let E and E' be the points where the tangent is parallel to BC and AC J respectively. We shall first compare the area of the triangle BEC, for instance, with that of the triangle ABC. Draw the tangent ET, which cuts CD at T. Draw the diam- eter EF, which cuts CB Sit F; and, finally, draw EK and FH parallel to the chord AB. By an elementary property of the parabola TC = CK. Moreover, CT =:EF = KH, and hence EF=CH/2=CD/4:. The areas of the two triangles BCE and BCD, since they have the •ame base BC, are to each other as their altitudes, or as EF is to CD. Henoe the area of the triangle BCE is one fourth the area of the triangle BCD, or one eighth of the area ^^ of the triangle ABC. The area of the triangle A CE' is evidently the same. Carrying out the tame process upon eacH of the chords BE, CE, CE', E'A, we 134 Fio.8 IV,§d5j SPECIAL METHODS 135 obtain four new triangles, the area of each of which is 5/8*, and so forth. The nth operation gives rise to 2" triangles, each having the area S/S*. The area of the segment of the parabola is evidently the limit approached by the sum of the areas of all these triangles as n increases indefinitely ; that is, the sum of the following descend- ing geometrical progression : S S -f-- and this sum is 4 S/3. It follows that the required area ijt equal to two thirds of the area of a parallelogram whose sides are AB and CD. Although this method possesses admirable ingenuity, it must be ailmitt*jd that its success dejxinds essentially upon certain special properties of the parabola, and that it is lacking in generality. The other examples of quadratures which we might quote from ancient writers would only go to corroborate this remark : each new curve required some new device. But whatever the device, the area to be evaluated was always split up into elements the number of which was made to increase indefinitely, and it was necessary to evaluate the limit of the sum of these partial areas. Omitting any further particular cases,* we will proceed at once to give a general method of subdivision, which will lead us naturally to the Integral Calculus. 65. General method. For the sake of definiteness, let us try to evaluate the area S bounded by a curvilinear arc AM By an axis xx' which does not cut that arc, and two perpendiculars ^.lo a-^d BB^ let fall upon xx' from y the points .^1 and B. We will suppose further that a par- allel to these lines AA^y BBq cannot cut the arc in more than one point, as indicated in Fig. 9. Let us divide the segment A^Bq into a certain number of etjual or unequal parts by the points P|, Pj, •••, P,_i, and through these points let us draw lines PiQi, I\Qt^ •••, P«-iQ«_i parallel to AA^ and meeting the arc AB in the points Qiy Q„ •••, Q,_„ respectively. x'o ^^ Rt fti A, />, A /Vi5» FiQ. 9 * A lar^o number of examples of determinations of areas, arcs, and Tolomas by the methods of ancient writers are to be found in Duhamel's TraUi, 186 DEFINITE INTEGRALS [IV, §65 Now draw through A a line parallel to xx', cutting Pi Qi at q^ ; through Qi a parallel to xx', cutting P^Q^ at qij and so on. We obtain in this way a sequence of rectangles Ri, R^, • • • , /?„ • • • , /?„. Each of these rectangles may lie entirely inside the contour ABBqAq, but some of them may lie partially outside that contour, as is indicated in the figure. Let a< denote the area of the rectangle Ri, and )S, the area bounded by the contour Pi.xPiQiQi-i- In the first place, each of the ratios fii/aif pt/aty •••, Pi/ocij ••• approaches unity as the number of points of division increases indefinitely, if at the same time each of the distances A^Pu P1P2, ■" , Pt-iPi, •• approaches zero. For the ratio ft/a,, for example, evidently lies between li/Pi_iQi_i and L{/P{,iQi_i, where Z,- and Z,- are respectively the minimum and the maximum distances from a point of the arc Q,- _ 1 Q, to the axis xx'. But it is clear that these two fractions each approach unity as the distance Pi^xPi approaches zero. It therefore follows that the ratio A + A + --- + ft which lies between the largest and the least of the ratios cti/fSi, ^i/fiit •'} ^n/Pni will also approach unity as the number of the rectangles is thus indefinitely increased. But the denominator of this ratio is constant and is equal to the required area S. Hence this area is also equal to the limit of the sum «! + org H + ^n> 3,s the number of rectangles n is indefinitely increased in the manner specified above. In order to deduce from this result an analytical expression for the area, let the curve AB be referred to a system of rectangular axes, the x axis Ox coinciding with xx', and let y =/(«) be the equation of the curve AB. The function /(a:) is, by hypothesis, a continuous function of x between the limits a and b, the abscissae of the points A and B. Denoting hy x^, x^, ■ - • , x^_i the abscissae of the points of division Pj, P,, -.., P,._i, the bases of the above rectangles are «i - a, x, - sci, . . . , jb^ - a;.._i, • . • , i - x„_^, and their altitudes are, in like manner, f{a),f(x,), ..., f{x,_,), ••., f{x^_,). Henoe the area S is equal to the limit of the following sum : (1) (x» - a)/(a) + (x, - x,)f{x,) + • • • + (6 - a:,_0/(a:„_0, M the number n increases indefinitely in such a way that each of the differences a^ ~ a, a-j - a?i, ... approaches zero. IV. {t»] 8P£C1AL MKTUODb 187 66. Examples. If the base AB be divided into n equal parts, each of length h {b ^ a = nh), all the rectangles have the same base h, and their altitudes are, respectively, /(«), /(« + A), /(a + 2 A), . . ., /[a + (n - 1) A]. It only remains to find the limit of the sum A }/(«) -f /(a + h) +f(a + 2 A) -h • • • +/[a + (n - 1)A] j, where as the integer n increases indefinitely. This calculation becomes easy if we know how to find the sum of a set of values /(x) corre- sponding to a set of values of x which form an arithmetic progres- sion; such is the case if f(x) is simply an integral power of x, or, again, iff(x)= sin7rtx or/(x)= coswix, etc. Let us reconsider, for example, the parabola x* = 2pi/, and let us try to find the area enclosed by an arc OA of this parabola, the axis of x, and the straight line x = a which passes through the extremity A. The length being divided into n equal parts of length A (nA = a), we must try to find by the above the limit of the sum 2^[A» + 4A« + . .. + («-!)• A']=g[l +4 + 9 + ■•. +(»-!)•]. The quantity inside the parenthesis is the sum of the squares of the first (n — 1) integers, that is, 7i(n — 1) (2 n — l)/6; and hence the foregoing sum is equal to n(n-l)(2n~l) 12 pn* As n increases indefinitely this sum evidently approaches the limit a*/6p = (l/3){a. a^/2p)y or one third of the rectangle constructed upon the two coordinates of the point Ay which is in harmony with the result found above. In other cases, as in the following example, which is due to Format, it is better to choose as points of division points whose abscissae are in geometric progression. Let us try to find the area enclosed by the curve y ^ Ax»^, the axis of X, and the two straight lines x a a, x = 6 (0 < a < 6), where 188 ' DEFINITE INTEGRALS [IV, §66 the exponent fi is arbitrary. In order to do so let us insert between a and *, n — 1 geometric means so as to obtain the sequence a, a(l + a), a(l + «)S -", «(1 + «)""'> *> where the number a satisfies the condition a (1 + «)" = *• Tak- ing this set of numbers as the abscissae of the points of division, the corresponding ordinates have, respectively, the following values : and the area of the ^th rectangle is Hence the sum of the areas of all the rectangles is Aa'^'^'all + (1 + ay + ' + (1 + a)»f'* + i)+ •••+(! + «y»-i)<'^ + ^>]. If ^ -f 1 is not zero, as we shall suppose first, the sum inside the parenthesis is equal to or, replacing a (1 -f- a)* by b, the original sum may be written in the form a ^(JM + l_aM + l) (l-^ay + '-l As a approaches zero the quotient [(1 H- a)**"^^ — l]/a approaches as its limit the derivative of (1 + a^-^^ with respect to a for a = 0, that is, fi + l'y hence the required area is /* + ! If ;i = — 1, this calculation no longer applies. The sum of the areas of the inscribed rectangles is equal to nAa, and we have to find the limit of the product na where n and a are connected by the relation a(l + a)* = 6. From thifl it follows that na — log - z — r = log - ''alog(l -Ha) *a log(l + ay IV.JflT] sprrrvT. ^fETHODS 189 where the symbol log den approaches zero, (1 4- a;'/* a| uct na approaches log (6 /a). A\og(b/a). "' (perian logarithm. As a i<* number «, aiid the prod- Hence the required area U equal to Fio. 10 67. Primitive functions. The invention of the Integral Calculus reduced the proljltiii of evaluating a plane area to the problem of finding a function whose derivative is known. Let y =/(af) be the equation of a curve referred to two rectangular axes, where the function f(x) is continuous. Let us consider the area enclosed by this curve, the axis of ar, a fixed ordinate 3/o/*o> and a variable ordinate MP, as a function of the abscissa x of the variable ordinate. In order to include all pos- sible cases let us agree to denote by A the sum of the areas enclosed by the given curve, the x axis, and the straight lines MqPoj ^^f*f each of the portions of this area being affected by a certain sign: the sign -f- for the portions to the right of MqPq and above Ox, the sign — for the portions to the right of 3/yPo and below Ox, and the opposite convention for por- tions to the left of MqJ^q. Thus, if MP were in the position M'P', we would take A equal to the difference MoPoC - M'P'C; and likewise, if MP were at M"P", A = M"P"D - MoPoD. With these conventions we shall now show that the derivative of the continuous function A, defined in this way, is precisely f(x). As in the figure, let us take two neighboring ordinates MP, XQ, whose abscissae are x and x -f- Ax. The increment of the area A.4 evidently lies between the areas of the two rectangles which have the same base PQ, and whose altitudes are, respectively, the greatest and the least ordinates of the arc MN. Denoting the maximum ordinate by // and the minimum by A, we may therefore write AAx < A^ < ^Ax, or, dividing by Aj, h < A.I /Ax < //. As Ax approaches zero, FT and h aj)pruiu.'h the same limit MP, or /(x), since /(x) is continuous. 140 DEFINITE INTEGRALS [IV, §G8 Hence the derivative of A is /(a;). The proof that the same result holds for any position of the point M is left to the reader. If we already know a primitive function oif{x), that is, a function F(x) whose derivative i8/(x), the difference A — F(x) is a constant, since its derivative is zero (§ 8). In order to determine this con- stant, we need only notice that the area A is zero for the abscissa a; = a of the line MP. Hence A =F(x)-F(a). It follows from the above reasoning, first, that the determination of a plane area may be reduced to the discovery of a primitive func- tion; and, secondly (and this is of far greater importance for us), that every continuous function f(x) is the derivative of some other function. This fundamental theorem is proved here by means of a somewhat vague geometrical concept, — that of the area under a plane curve. This demonstration was regarded as satisfactory for a long time, but it can no longer be accepted. In order to have a stable foundation for the Integral Calculus it is imperative that this theo- rem should be given a purely analytic demonstration which does not rely upon any geometrical intuition whatever. In giving the above geometrical proof the motive was not wholly its historical interest, however, for it furnishes us with the essential analytic argument of the new proof. It is, in fact, the study of precisely such sums as (1) and sums of a slightly more general character which will be of preponderant importance. Before taking up this study we must first consider certain questions regarding the general properties of functions and in particular of continuous functions.* II. DEFINITE INTEGRALS ALLIED GEOMETRICAL CONCEPTS 68. Upper and lower limits. An assemblage of numbers is said to have an upper limit (see ftn., p. 91) if there exists a number N so large that no member of the assemblage exceeds N. Likewise, an assemblage is said to have a lower limit if a number N' exists than which no member of the assemblage is smaller. Thus the assem- blage of all positive integers has a lower limit, but no upper limit j • Among the most important works on the general notion of the definite integral tlMM ■hould be mentioned the memoir by Rieraann : Vher die Mdglichkeit, eine Func- tion dureh eine trigonometrische Reihe darzustellen (Werke, 2d ed., Leipzig, 1892, p.SS0; and alio French translation by Ijiugel, p. 226) ; and the memoir by Dferboux, to wbleh we have already referrod : Sur lea fonctionn ilitcontinueft {Annuleti de VEcole Normals Bup^rieure, 2d seriMt, Vol. IV). IV,§.wj ALLIED GEOMETRICAL CONCEPTS 141 the assemblage of all integers, positive aiid negatiye, has neither; and the assemblage of all rational numbers between and 1 has both a lower and an upper limit. Let (E) be an assemblage which has an upper limit With respect to this assemblage all numbers may be divided into two classes. We shall say that a nunilxT a belongs to the first class if there are memlx*rs of the assemblage (E) which are greater than a, and that it belongs to the second class if there is no member of the assemblage (/;) greater than a. Since the assemblage (E) has an upper limit, it is clear that numbers of each class exist. If A he a number of the first class and B a number of the second class, it is evident that A <^; there exist members of the assemblage (E) which lie between A and B^ but there is no member of the assem- blage (E) which is greater than B. The number C = (v4 -f B)/2 may belong to the first or to the second class. In the former case we should replace the interval (A, B) by the interval (C, /i), in the latter case by the interval (/I, C). The new interval (Ai^ B^) is half the interval (.1, B) and has the same properties : there exists at least one member of the assemblage (A') which is greater than .-li, but none which is greater than Bi. Operating upon (.4i, Bi) in the same way that we operated upon (A, B), and so on indefinitely, we obtain an unlimited sequence of intervals (.1, J5), (Ai, Bi), (/I,, 5j), •••, each of which is half the preceding and possesses the same property as (.1, B) with respect to the assemblage (E). Since the numbers .1, A I, vlj, •••, A^ never decrease and are always less than iJ, they approach a limit X (§ 1). Likewise, since the numl)ers By B^ 5,, • • never increase and are always greater than A , they approach a limit X'. Moreover, since the difference B^ — .1, =(^B — -4)/2" approaches zero as n increases indefinitely, these limits must be equal, i.e. X' = X. Let L be this common limit; tlien L is called the vjiper limit of the assemblage (A'). From the manner in which we have obtained it, it is clear that L has the following two properties : 1) No tnemher of the assemblage (E) w greater than L. 2) There alirat/s exists a member of the assembla/je (E) which is ijreater than Z, — e, where c is any arbitrarily small positive number. For let us suppose that there were a member of the assemblage greater than Z., say L + A (A > 0). Since B^ approaches L as n increases indefinitely, B^ will be less than L -^ h after a certain value of 71. But this is imjx>ssible since B^ is of the second class. On the other hand, let c be any positive numl)er. Then, after a 142 DEFINITE INTEGRALS [IV, §69 certain value of n, i4. will be greater than i — € ; and since there are members of (E) greater than A^, these numbers will also be greater tlum L — €. It is evident that the two properties stated above can- not apply to any other number than L. The upper limit may or may not belong to the assemblage (E). In the assemblage of all rational numbers which do not exceed 2, for instance, the number 2 is precisely the upper limit, and it belongs to the assemblage. On the other hand, the assemblage of all irra- tional numbers which do not exceed 2 has the upper limit 2, but this upper limit is not a member of the assemblage. It should be particularly noted that if the upper limit L does not belong to the assemblage, there are always an infinite number of members of (E) which are greater than L — c, no matter how small c be taken. For if there were only a finite number, the upper limit would be the largest of these and not L. When the assemblage consists of n different numbers the upper limit is simply the largest of these n numbers. It may be shown in like manner that there exists a number L\ in case the assemblage has a lower limit, which has the following two properties : 1) No member of the assemhlage is less than L'. 2) There exists a member of the assemblage which is less than V 4- c, where t is an arbitrary positive number.* This number Z' is called the lower limit of the assemblage. 69. Oscillation. Let f(x) be a function of x defined in the closed f interval (a, b) ; that is, to each value of x between a and b and to each of the limits a and b themselves there corresponds a uniquely deter- mined value of /(x). The function is said to he finite in this closed interval if all the values which it assumes lie between two fixed numbers A and B. Then the assemblage of values of the function has an upper and a lower limit. Let M and m be the upper and lower limits of this assemblage, respectively ; then the difference •Whenerer ail numbers can be separated into two classes A and B, according to any charactarlitic property, in such a way that any number of the class .4 is less than any number of the claM B, the upper limit L of the numbers of the class A is at the mm» time the lower limit of the numbers of the class B. It is clear, first of all, that any nttmbw greater than L l)elong8 to the class B. And if there were a number L'<L ; to thaolaM B, then every number greater than // would belong to the class B, &twj numb«r lew than L belongs to the class A, every number greater than L balonfi to the claas B, and L itself may belong to either of the two classes. t Tha word " doMd " to used merely for emphasis. See § 2. — Trans. IV, §70] ALLIED GEOMETRICAL CONCEPTS 143 A s= Af — m is called the oscillation of the function /(x) in the interval (a, b). These definitions lead to sevcial ifinarks. In order that a funo- tion be finite in a closed interval i^u, b) it is not sufficient that it should have a finite value for every value of x. Thus the function defined in the closed interval (0, 1) as follows : /(O) = 0, /(x) = 1/x for X > 0, has a finite value for each value of x ; but nevertheless it is not finite in the sense in which we have defined the word, for/(x) > A if we take x<l/A. Again, a function which is finite in the closed interval (a, b) may take on values which differ as little as we please from the upper limit M or from the lower limit m and still never assume these values themselves. For instance, the function /(x), defined in the closed interval (0, 1) by the relations /(0) = 0, f(x) = l-x for 0<x<l, has the upper limit M = 1, but never reaches that limit. 70. Properties of continuous functions. We shall now turn to the study of continuous functions in particular. Theorem A. Letf(x) be a function which is continunus ui mr nosed interval (a, b) and € an arbitrary positive number. Then we can always break up the interval (a, b) into a certain number of partial intervals in such a way that for any two' values of the variable whatever^ a:' a7id x", which belong to the same partial interval^ we always have |/(x')— /(x")| < c. Suppose that this were not true. Then let c=(a-|-d)/2; at least one of the intervals (a, c), (c, b) would have the same prop- erty as (rt, 6); that is, it would be impossible to break it up into partial intervals which would satisfy the statement of the theorem. Substituting it for the given interval (a, b) and carrying out the reasoning as above (§ 68), we could form an infinite sequence of intervals (a, ft), (oj, ft,), (a„ 6,), • •, each of which is half the preced- ing and has the same property as the original interval (a, b). For any value of n we could always find in the interval (o., b^) two numbers x' and x" such that (/(x')— /(x")| would be larger than c. Now let X be the common limit of the two sequences of numbers a, ai, aj, • •• and b, bi, /^,, • • •. Since the function /(j-) is continuous for X = X, we can find a number i; such that |/(x)— /(X)| <€/2 144 DEFINITE INTEGRALS [IV, §70 wheneyer |x - A| is less than 17. Let us choose n so large that both a, and d, differ from k by less than rj. Then the interval (a„, K) will lie wholly within the interval (X - 1;, X + ^7) ; and if x' and x" are any two values whatever in the interval (a„, 6„), we must have \f(x') -/(X) I < c/2, \f(x").-f(\) I < e/2, and hence l/Cx*) -/(x") | < e. It follows that the hypothesis made above leads to a contradiction ; hence the theorem is proved. Corollary L Let a, Xj, ar„ • • • , x^.i, i be a method of subdivision of the interval (a, h) into p subintervals, which satisfies the con- ditions of the theorem. In the interval (a, x^ we shall have |/(x) I < |/(a) I + e ; and, in particular, \f(xy) \ < \f{a) \ + c. Like- wise, in the interval (xi, x^ we shall have |/(ic)| < |/(a:i) | -f- c, and, a fortioriy \f(x) \ < \f(a) | 4- 2 c ; in particular, for x = x^, |/(^) I < |/(") I + 2 € ; and so forth. For the last interval we shall have l/WI< 1/(^,-01 + '<!/« I +i'«- Hence the absolute value of f(x) in the interval (a, b) always remains less than |/(a) 1 4- p^. It follows that every function which is continuous in a closed interval (a, b) is finite in that interval. Corollary II. Let us suppose the interval (a, b) split up intojo sub- intervals (a, Xx)j (xj, Xj), • • • , (Xp_i, b) such that |/(x') —f(x")\< c/2 for any two values of x which belong to the same closed subinterval. Let 17 be a positive number less than any of the differences x^ — a, X, — Xj • • • , 6 — Xp_i. Then let us take any two numbers whatever in the interval (a, b) for which |x' — x"| < 77, and let us try to find an upper limit for |/(x') — /(x")|. If the two numbers x' and x" fall in the same subinterval, we shall have |/(x') — /(x")|<c/2. If they do not, x' and x" must lie in two consecutive intervals, and it is easy to see that |/(x') -f(x") \ < 2 (e/2) = e. Hence cor- responding to any positive member e another positive number -q can be found such that |/(x')-/(x")|<., where x' and x" are any two numbers of the interval (a, b) for which Ix* — »"|<iy. This property is also expressed by saying that the function f(x) is uniformly continuous in the interval (a, b). Thborbm B. a function f(x) which is continuous in a closed intemal (a, b) takes on every value between f(a) and f(b) at least once for some value of x which lies between a and b. IV, §70] ALLIED GEOMETRICAL CONCEPTS 146 Let us first consider a particular case. Suppose that /(a) and f{b) have opposite signs, — that /(a) < and /(A) > 0, for instance. We shall then show that thero exists at least one value of x between a and b for which f{x) = 0. Now/(3r) is negative near a and posi- tive near b. Let us consider the assemblage of values of x between a and b for which f{x) is positive, and let A be the lower limit of this assemblage (a < X < b). By the very definition of a lower limit /(A — h) is negative or zero for every positive value of h. Hence /(A), which is the limit of /(A — A), is also negative or zero. But /(A) cannot be negative. For suppose that /(A) = — m, where m is a positive number. Since the function f(x) is continuous for X = A, a number rj can be found such that \f(x) — /(A)| < m when- ever |a; — A| < 17, and the function f(x) would be negative for all values of x between A and A -h 17. Hence A could not be the lower limit of the values of x for which /(x) is positive. Consequently /(X) = 0. Now let N be any number between f(a) and f(b). Then the function <^(x) =f(x) — N is continuous and has opposite signs for X = a and x = b. Hence, by the particular case just treated, it vanishes at least once in the interval (a, b). Theorem C. Every function which w continuous in a closed inter- val (a, b) actnally assumes the value of its upper and of its lower limit at lea^t once. In the first place, every continuous function, since we have already proved that it is finite, has an upper limit M arid a lower limit m. Let us show, for instance, that /(x) = 3/ for at least one value of xin the interval (<«, b). Taking c = (a •\- b)/2, the upper limit of f(x) is equal to M for at least one of the intervals (a, c), (<•, 6). Let us replace (a, b) by this new interval, repeat tlie process upon it, and so forth. Reasoning as we have already done several times, we could form an infinite sequence of intervals (a, b)y (a^, 6,), (a,, A,), ••., each of which is half the preceding and in each of which the upper limit of /(x) is M. Then, if A is the cx)mmon limit of the sequences a, Ci, • • • , a„ • • • and A, ij, • • • , ft,, • • • , /(A) is equal to M. For suppose that /(A) = M — hf where h is positive. We can find a positive number rj such that/(x) remains between /(A) 4- A/ 2 and /(A) — A/ 2, and therefore less than A/— A/2 as long as x remains between A — 17 and A -I- 17. Let us now choose n so great that a„ and ft, differ from their common limit A by less than rj. Then the interval (a,, ft,) lies 146 DEFINITE INTEGRALS [IV, §71 wholly inside the interval (\ — rj,\ + -q), and it follows at once that the upper limit of f{x) in the interval (a„, b^ could not be equal to 3/. Combining this theorem with the preceding, we see that any func- tion which is continuous in a closed interval (a, h) assumes, at least oneey every value between its upper and its lower limit. Moreover theorem A may be stated as follows : Given a function which is continuous in a closed interval (a, b), it is possible to divide the inter- val into such small subregions that the oscillation of the function in any one of them will be less than an arbitrarily assigned positive number. For the oscillation of a continuous function is equal to the difference of the values of /(a) for two particular values of the variable. 71. The sums S and s. Let f(x) be a finite function, continuous or discontinuous, in the interval (a, b), where a<b. Let us sup- pose the interval (a, b) divided into a number of smaller partial intervals (a, Xi), (xi, Xg), •••, (ajp-i, b), where each of the numbers a; J, a;,, • • •, Xp.j is greater than the preceding. Let M and m be the limits of f{x) in the original interval, and M^ and m, the limits in the interval («,_!, Xf), and let us set S = M^(x^ - a) 4- M,(x^ -x^)^--- + M^(b - x^_,), s = mi (Xi — a) -{- m2(x2 — x{) -] h m^ (b — iCp_i). To every method of division of (a, b) into smaller intervals there corresponds a sum S and a smaller sum s. It is evident that none of the sums S are less than m(b — a), for none of the numbers Af^ are less than m ; hence these sums S have a lower limit /.* Like- wise, the sums «, none of which exceed M(b — a) have an upper limit r. We proceed to show that /' is at most equal to I. For this purpose it is evidently sufficient to show that s<S' and s' ^ .S", where 5, s and 5', s' are the two sets of sums which correspond to any two given methods of subdivision of the interval (a, b). In the first place, let us suppose each of the subintervals (a, Xj), ('i» *,), ••• redivided into still smaller intervals by new points of division and let <h Vu y»f "i t/k-it a^i, yjt+i, •., yj_i, Xj, 2// + i, ••, b • ll/(«) itaeoiuiUnt, 8=a,M=:m, and, in general, all the inequalities mentioned equAtiou. — TaANi. IV, §72] ALLIED GEOMETRICAL CONCEPTS UJ be the new suite thus obtained. This new method of subdivision is called consecutive to the first. Let S and <r denote the sums anal- ogous to .S' and s with respect to this new method of division of the interval (a, b), and let us compare S and « with 2 and o-. Let us compare, for example, the portions of the two sums S and 2 which arise from the interval (a, Xi). Let M[ and m[ be the limits of f(x) in the interval (a, y,), 3/,' and m^ the limits in the interval (i/i) yi)i ' ■ ) ^^k ^^^ W the limits in the interval (jft-u *i)* Then the portion of S which comes from (a, x^ is M{{y, - «) + A/Ky« - y,) + • • + 3/i(ar, - y,_0 ; and since the numbers 3/|, A/j, •• •, 3// cannot exceed 3/,, it is clear that the above sum is at most equal to 3/i (a*i — a). Likewise, the portion of 2 which arises from the interval (x„ a*,) is at most equal to 3/,(a'j — a-,), and so on. Adding all these inequalities, we find that 2 ^ ^y and it is easy to show in like manner that <r ^ ». Let us now consider any two methods of subdivision whatever, and let Sy s and S\ s' be the corresponding sums. Superimposing the points of division of these two methods of sulxlivision, we get a third method of subdivision, which may be considered as consecu- tive to either of the two given methods. Let 2 and <r be the sums with respect to this auxiliary division. By the above we have the relations 2<5, «r>«, 2<.S% (r>»'; and, since 2 is not less than o-, it follows that a'< 5 and 8<S', Since none of the sums S are less than any of the sums «, the limit / cannot be. less than the limit /'; that is, /^/'. 72. Integrable functions. A function which is finite in an inter- val (a, b) is said to l>e mtegrable in that interval if the two sums S and s approac^h the same limit when the number of the partial intervals is indefinitely increased in such a way that eacli ^f t>iocp partial intervals approaches zero. The necessary and sufficient condition that a function he integrable in an interval is that corresponding to any positive number c another nuviber rj exists sttch that S — s is less than c whenever each of the partial intervals is less than rj. This condition is, first, necessary, for if &' and s have the same limit /, we can find a number ij so small that |5 ~ /"I wd |« — /| are 148 DEFINITE INTEGRALS [IV, §72 each less than c/2 whenever each of the partial intervals is less than iy. Then, a fortiori, S — sis less than c. Moreover the condition is sufficient, for we may write * S-s = S-I + I-I' + I'-s, and since none of the numbers S - I, I - I', I' - s can be negative, each of them must be less than c if their sum is to be less than c. But since / — /' is a fixed number and c is an arbitrary positive number, it follows that we must have /' = /. Moreover S — I<€ and I — s<€ whenever each of the partial intervals is less than rj, which is equivalent to saying that S and s have the same limit /. The function f(x) is then said to be integrahle in the interval (a, b), and the limit / is called a definite integral. It is represented by the symbol I=^Jf(x)dx, which suggests its origin, and which is read ''the definite integral from a to 6 of f(x) dx.'' By its very definition / always lies between the two sums S and s for any method of subdivision whatever. If any number between S and s be taken as an approximate value of /, the error never exceeds S — s. Every continuous function is integrahle. The difference 5 — s is less than or equal to (h — a)ui, where a» denotes the upper limit of the oscillation of f(x) in the partial intervals. But ri may be so chosen that the oscillation is less than a preassigned positive number in any interval less than r} (§ 70). If Uien i; be so chosen that the oscillation is less than e/(b — a), the difference S — s will be less than «. Any monotonically increasing or monotonically decreasing function in an interval is integrahle in that interval^ A function /(x) is said to increase rnonotonically in a given interval (a, b) if for any two values x\ x" in that interval f(x') ^.f(x") when- ever x' > x". The function may be constant in certain portions of the interval, but if it is not constant it must increase with x. Dividing the interval (a, b) into n subintervals, each less than rj, we may write S =/(xO (X, - a) +/(x,) (X, - xO + • • • +/(ft) (b - x_0, M ^f{a)(x, - a) +/(xO(x. - xO -\""+f(x,,,)(b - x,.,), •For Um proof that / and r exist, see §73, which may be read before § 72. —Trans. 1V,J721 ALLIED GEOMETRICAL CONXEPTS 149 for the upper limit of f(x) in the interval (a, arj), for instance, is precisely /(x^), the lower limit /(a); and so on for the other subintervals. Hence, subtracting, i- - * = (X, - o)[/(x,)-/(a).l + (X. - x,)[/(x.) -/(x,)] + •• -l-(*-x...)[/(«)-/(x._,)]. None of the differences which occur in the right-hand side of this equation are negative, and all of the differences Xi — o, x^ — x^ '■• are less than 17; consequently S-»< 1) [/('.) -/(«) +/{'.) -/(^.) + • • • +/(i) - /(=".-.)]. or 5-,<,[/(i)-/(a)], and we need only take in order to make S — s< t. The reasoning is the same for a mono- tonically decreasing function. Let us return to the general case. In the definition of the inte- gral the sums S and s may be replaced by more general expres- sions. Given any method of subdivision of the interval (a, b) : a, a:i, a;,, • •, ar,_,, ar,, ••., x^_^, b; let ^1, ^2» • • , ^,, • • • be values belonging to these intervals in order (a-,_, < $i < Xi). Then the sum (2)|X/(ft)(^^-^.-i) = I' V(^i)(^i - «) +/(«(^« - xi) + • • • +f($nKb - *.-,) evidently lies between the sums S and », for we always have 7/1, </(^,) < 3/,. If the function is integrable, this new sum has the limit /. In particular, if we suppose that ^1, ^„ • • , ^, coincide with o, Xj, ••., «,_„ respectively, the sum (2) reduces to the sum (1) considered above (§ C>5). There are several propositions which result immediately from the definition of the integral. We have supposed that a < ft ; if we now interchange these two limits a and 6, each of the factors X| — «,_, changes sign ; hence J j\x)dx=.-jyix)dx. 150 DEFINITE INTEGRALS [IV, §72 It also evidently follows from the definition that rf(x)dx = ff(x)dx 4- ff(x)dx, %/a *Ja Jc at least if c lies between a and h\ the same formula still holds when h lies between a and c, for instance, provided that the function f{x) is integrable between a and c, for it may be written in the form ff(x)dx = ff(x)dx - ff(x)dx = ff(x)dx + rf(x)dx. If f{x) = A<l>(x) + B\l/(x), where A and B are any two constants, we have fix)dx = A I <f>(x)dx-\- B I xlf(x)dx, and a similar formula holds for the sum of any number of functions. The expression /(^,) in (2) may be replaced by a still more gen- eral expression. The interval (a, h) being divided into n sub- intervals (a, Xi), • • • , (x,_i, a;,), • • • , let us associate with each of the subintervals a quantity ^„ which approaches zero with the length 05, — x,_i of the subinterval in question. We shall say that ^^ approaches zero uniformly if corresponding to every positive num- ber c another positive number t^ can be found independent of i and such that l^;^ I < c whenever x^ — Xi_^ is less than iy. We shall now proceed to show that the sum «'=X[/(^i-.)+ «('».■-»=.•-.) (=1 approaches the definite integral ]^f(x)dx as its limit provided that ^, approaches zero uniformly. For suppose that 17 is a number 80 small that the two inequalities X/(^<-i)(aJ.-aj,-0- ^f{x)dx <e, 1^,1 <C are satisfied whenever each of the subintervals ic» — a;,._i is less than 17. Then we may write '-£' /{x)dx = [X/(»<-i)(a'< - 'i-x) -£nx)dx'\ + ^ i,{x, - x,_o, IV. 573] ALLIED GEOMKTKICAL CONCEPTS 151 and it is clear that we shall have -f. f{x)dx <€ + «(*-«) whenever each of the subintervals is less than rj. Thus the theorem is proved.* 78. Darboux't theorem. Given any function /(z) which \& finite in an inter- val (a, b)\ tlie sums S and s approach their limitii / and /% respectively, when tlie number of subintervals increases indefinitely in such a way that each of them approachee zero. Let ua prove this for the sum »S', for instance. We shall suppose that a<6, and that/(z) is positive in the interval (a, 6), which can be brought about by adding a suitable constant to/(x), which, in turn, amounts to adding a constant to eacli of the sums S. Then, since the number / is the lower limit of all the sums ^, we can find a particuhir method of subdivision, say for which the sum S is less than I + f/2, where e is a preassigned positive num- ber. Let us now consider a division of (a, b) into intervals less than ij, and let us try to find an upper limit of the corresponding sum S'. Taking first those inter- vals which do not include any of the points Zi, x^, • • •, 2p_i, and recalling the reasoning of § 71, it is clear that the portion of iS" which cumes from these inter- vals will be less than the original sum S, that is, less than / -f e/2. On the other hand, the number of intervals which include a point of the set Zj, Zt, • • • , Xp-\ cannot exceed p — 1, and hence their contribution to the sum S' cannot exceed (p — \)Mri, where if is the upper limit of /(z). Hence S'<7-|-e/2 + (p-l)Jfi», and we need only choose ij less than e/2 3f (p - 1) in order to make S' less than / + «. Hence the lower limit / of all the sums S is also the limit of any sequence of 5's which corresponds to uniformly infinitesimal subintervals. It may be shown in a similar manner that the sums a have the limit /'. If the function /(z) is any function whatever, these two limits I and /' are in general different. In order that the function be integrable it is necessary and sufficient that /' = I. 74. First law of the mean for integrals. From now on we shall assume, unless something is explicitly said to the contrary, that the functions under the integral sign are continuous. * The al)ove theorem can be extended without difficulty to double and triple inte- gralH ; we shall make use of it in several places ($§ 8i), 96, 97, 131, 144, etc.). The pniposition is essentially only an application of a theorem of Dobamel's at^*ortiinK to which the limit of a sum of inHnitesimals remains unchanged when each of tht> infiiiitcsimalH is replactnl hy another infinitesimal which differs from the given JntiniteHinml by an iiitinittwinml of higher order. (See an article hy W. P. Osgtxxl, AnnaU of Matheuiatic*, '31 series, Vol. IV, pp. 161-178: Th9 Inteffral <u the JAinit of a Sum ami a Thtoretn o/ Duhamel'a.) 152 DEFINITE INTEGRALS [IV, §74 Let/(x) and ^(x) be two functions which are each continuous in the interval (a, b), one of which, say <t>(^), has the same sign throughout the interval. And we shall suppose further, for the sake of definiteness, that a<b and <f> (x) > 0. Suppose the interval (a, b) divided into subintervals, .and let i it •••» A» ••• ^ values of x which belong to each of these smaller intervals in order. All the quantities f(i,) lie between the limits M and m of /(x) in the interval (a, b) : m<f(i,)<M. Let us multiply each of these inequalities by the factors respectively, which are all positive by hypothesis, and then add them together. The sum ^f($^)<|>($i)(Xi-x^_^) evidently lies between the two sums w2<^(^,) (x.- - x,._i) and M2</)(^.) (x,- - x,_,). Hence, as the number of subintervals increases indefinitely, we have, in the limit, / f <f>(x)dx^ I f(x)<ji(x)dx<M j <l>(x)dx, a J a Jo- which may be written fix) <t>(x)dx = ix j <f>(x) dxj where /i lies between m and M. Since the function f{x) is con- tinuous, it assumes the value /i for some value ^ of the variable which lies between a and b ; and hence we may write the preceding equation in the form (3) rf(x)<t>(x)dx =/(^) C\{x)dx, %Ja %J a where i lies between a and b. * If, in particular, <^ (x) = 1 , the integral ^ dx reduces to {b — a) by the very definition of an inte- gral| and the formula becomes w £f(x)dx=(b «)/(«)• • Tha lower ilgn hoIdH in the preceding relations only when/(i) = k. It is evidenc that iIm) furmula «iUl holdit, however, and that a<( < & in any case. —Trans. IV, § 75] ALLIED GEOMETRICAL CONCEPTS 154 76. Second law of tho mean for integrals. There is a second formula, due to Bonnet, which he deduced from an Important lemma of AbePs. Lemma. Let co, <i, • • ,9pbeaHiof monoUmicaUy decrmutng pogUive quamU- ties^ anduo,ui,- • ,UpOie »ame number of arbitrary positive or negative quaniUiee. If A and B are rtapectively the greatest and tke leatt qf cUl cfthe nune fo ss ««, s, = uo -F uj . • • • , 1^ = Wo + ui + • • • + «p, <A« turn 8 = «oiio + nwi + • • • + fpH^ will lie between Ato and JUo, i.e. Ato ^ >' ^ IUq. For we have whence the sum S is equal to «o(<o - <i) + ai (<i - <j) + • -f ap-i (<p-i - «p) + tp€p. Since none of the differences <o — <i, <i — <«, • • •, <p-i — «i» are negatiye, two limits for S are given by replacing Soi Si , • • • , «p by their upper limit A and then by their lower limit B. In this way we find 8<A{«o-€i-\-ei-et + \- ep_i - ep -f ep) = Ate, and it is likewise evident that »S' ^ Be©. Now let/(x) and ^(x) be two continuous functions of z, one of which, ^(«), is a positive monotonically decreasing function in the interval a<x<b. Then tlje integral f^f{x)<p(x)dx is the limit of the sum /(a)0(a)(xi - a) +/(zi)0(xi)(xa - xi) + • • •• The numbers 0(a), 4>{xi), - • • form a set of monotonically decreasing positive numbers; hence the above sum, by the lemma, lies between A<p(a) and B<p{a)t where A and B are respectively the greatest and the least among the following sums: /(a)(xi-a), /(a) {xi - a) + /(xi) (X, - xi) , /(a) (xi - a) +/(xi) (X, - xi) + . . . +/(x,-i) (6 - «,_i). Passing to the limit, it is clear that the integral in question must lie between Ai^{a) and Bi(p((t), where Ai and Bi denote the maximum and the minimum, respectively, of the integral f^^/{x)d£, as c varies from a to 6. Since this inte- gral is evidently a continuous function of its upper limit c (§ 76), we may write the following formula : (6) j^/(x)0(x)dx = 0(a)j]/(x)dx, a<(<b. When the function ^(x) is a monotonically decreasing function, without being always positive, there exists a more general formula, due to Weientrasa. In such a case let us set ^ (x) = (b) + f (x). Then f (x) is a positiTe monotoo- ically decreasing function. Applying the formula (6) to it, we find jj(x)^{x)dx = [0(a) - 0(6)] j] /(x)dx. 154 DEFINITE INTEGRALS [IV, §76 From this it is easy to derive the formtda J /(x)0(x)dx =^ /(x)0(6)dx + [0(a) - 0(6)] j^ f{x)dx, or ^ y /(x)0(x)dx = if>{a)fjf{x)dx-]- <f>{b)f f{x)dx. SimilAr formulte exist for the case when the function 0(x) is increasing. 76. Return to primitive functions. We are now in a position to give a purely analytic proof of the fundamental existence theorem (S 67). hetJXx) be any continuous function. Then the definite integral ^<*>=X f(t)dt, where the limit a is regarded as fixed, is a function of the upper limit X. We proceed to show that the derivative of this function isf{x). In the first place, we have Xx + h f(t)dt, or, applying the first law of the mean (4), F{x^h)-F(x) = hf{t), where ^ lies between x and x -\- h. As /i approaches zero, /(^) approaches f{x) ; hence the derivative of the function F(x) is f{x), which was to be proved. All other functions which have this same derivative are given by adding an arbitrary constant C to F(x). There is one such function, and only one, which assumes a preassigned value yQ for « = a, namely, the function yo+jJ{t)dt. When there is no reason to fear ambiguity the same letter x is uaed to denote the upper limit and the variable of integration, and f'f(x)dx is written in place of fjf{t)dt. But it is evident that a definite integral depends only upon the limits of integration and the form of the function under the sign of integration. The letter which denotes the variable of integration is absolutely immaterial. Every function whose derivative is f(x) is called an indefinite inUffral of /(x), or a jyrimitive function of f{x)y and is represented by tiie symbol IV, fW] ALLIED GEOMETRICAL CONCEPTS 166 the limits not being indicated. By the above we evidently have Conversely, if a fuuction F{x) whose derivative is /(x) can be discovt^rt'd bv :iiiv iiietliod wliatever, we mav write i f{x)dx = F(ar)-H C. In order to determine the constant C we need only note that the left-hand side vanishes for x = a. Hence C = — /^(a), and the fundamental formula becomes (6) £/{x)dx = F(x)-F{a). If in this formula /(j-) be replaced by F'(ar), it becomes F(x)-F(a)=£FXx)dx, or, applying the first law of the mean for integrals, F(x)-F(a) = (x-a)F'(0, where $ lies between a and x. This constitutes a new proof of the law of the mean for derivatives ; but it is less general than the one given in section 8, for it is assumed here that the derivative F'(x) is continuous.' We shall consider in the next chapter the simpler classes of func- tions whose primitives are known. Just now we will merely state a few of those which are apparent at once : Ja(x - aydx = A ^^~2^l^' + ^> a -h 1 :9fc 0; / A — — = A log (x — a)'\- C; I cosxf/r =r sin J- 4- c'; j s'mxdx = ^ COBX -^ C; f eT'dx = — + r, m ^ Oj ]£6 DEFINITE INTEGRALS [IV, {76 rj^ = arctana + C; J;^^= = arosina! + C; J-^= = log(» + V?TT)+C, J=a^ = log/(x)+C. The proof of the fundamental formula (6) was based upon the assumption that the function f{x) was continuous in the closed inter- val (a, b). If this condition be disregarded, results may be obtained which are paradoxical. Taking f{x) = l/x% for instance, the for- mula (6) gives 'dx 1 1 b rdx_ Ja ^'~ The left-hand side of this equality has no meaning in our present system unless a and b have the same sign ; but the right-hand side has a perfectly determinate value, even when a and b have different signs. We shall find the explanation of this paradox later in the study of definite integrals taken between imaginary limits. Similarly, the formula (6) leads to the equation If /(a) and/(6) have opposite signs, /(x) vanishes between a and b, and neither side of the above equality has any meaning for us at present. We shall find later the signification which it is convenient to give them. Again, the formula (6) may lead to ambiguity. Thus, if f(x)^l/(l + x^), we find f dx arc tan b — arc tan a. Here the left-hand side is perfectly determinate, while the right- hand side has an infinite number of determinations. To avoid this ambiguity, let us consider the function '<-'-X'r^. This function F(x) is continuous in the whole interval and van- iihet with x. Let us denote by arc tan x, on the other hand, an angle between - 'w/2 and + 7r/2. These two functions have the IV, $77] ALLIED GEOMETRICAL CONCEPTS 167 same derivative and they both vanish for x = 0. It follows thai they are equal, and we may write the equality r' dx r'_dx_ rdx_ ^ . I r-— -5 = I ^ . , — I r-; — i = arc tan A - arc tan a, where the value to be assigned the arctangent always lies between -7r/2 and 4-7r/2. In a similar manner we may derive the formula /^ = arc sin b — arc sin a, VI — x' i: where the radical is to be taken positive, where a and b each lie l)etween — 1 and -f 1, and where arc sin x denotes an angle which lies between — 7r/2 and -f 7r/2. 77. Indices. In general, when the primitive F{x) is multiply determinate, we ghould choose one of the initial values F{a) and follow the continuous variation of this brancli as x varies from a to b. Let us consider, for instance, the integral J. p'+v 1 1 +/'(») where /(X) = ^ and where P and Q are two functions which are both continuous in the interval (a, b) and which do not both vanish at the same time. If Q does not vanish between a and 6, /(x) does not become infinite, and arc tan/(x) remains between — n/2 and + ic/2. But this is no longer true, in general, if the equation Q = has roots in this interval. In order to see how the formula must be modified, let us retain the Convention that arc tan signifies an angle between — ic/2 and -|- x/2, and let us suppose, in the first place, that Q vanishes just once between a and 6 for a value x = c. We may write the integral in the form r V(^)dx r^- . r'""'4. r* where e and e' are two very small positive numbers. Since /(«) does not become Infinite between a and c - c, nor between c -i- 1' and 6, this m«y again be written X 6 -^-— = arc tan/(c - c) - arc tAn/(a) + arc tan/(6) - arc t«n/(c + O + f Several cases may now present themselves. Suppose, for the sake of definite- ness, that /(x) becomes infinite by passing from + » to — eo. Then /(c — t) will be positive and very large, and arc tan/(c - c) will be very near to it/2 ; while 158 DEFINITE INTEGRALS [IV, § 78 tit + O w»» ^ negative and very large, and arc tan/(c + O will be very near - ir/2. Alao, the integral /^If will be very small in absolute value; and, to the limit, we obtain the formula X' £ Zi?)*L =^ + arctan/(6) - arctan/(a). f„ l+/«(x) Similarly, it is eaay to show that it would be necessary to s^traxt tc if /(x) pMsed from - oo to + 00. In the general case we would divide the interval (a, 6) into subintervals in such a way that /(x) would become infinite just once in each of them. Treating each of these subintervals in the above manner and adding the resulta obtamed, we should find the formula ' r(x)dg ^ arctan/(6) - arctan/(a) ^ {K - K') it, where K denotes the number of times that/(x) becomes infinite by passing from + 00 to - 00, and K' the number of times that f{x) passes from — co to + «. The number K - K' is called the index of the function /(x) between a and 6. When/(x) reduces to a rational function Vi/V, this index may be calculated by elementary processes without knowing the roots of V. It is clear that we may suppose Vi prime to and of less degree than F, for the removal of a poly- nomial does not affect the index. Let us then consider the series of divisions necessary to determine the greatest common divisor of Fand Fi, the sign of the remainder being changed each time. First, we would divide F by Fi, obtaining a quotient Qi and a remainder - Fa. Then we would divide Vi by F2, obtaining a quotient Q» and a remainder — Fs ; and so on. Finally we should obtain a con- stant remainder - F«+ 1. These operations give the following set of equations : F = FiQi - Fa, Vl =F2Q2-F8, F„_i = F„Q„-Fn+i. The sequence of polynomials (7) F, Fi, Fa, .-., Vr-u Fr, Fr+i, ..-, F„, F„+i has the essential characteristics of a Sturm sequence : 1) two consecutive poly- nomials of the sequence cannot vanish simultaneously, for if they did, it could be shown successively that this value of x would cause all the other polynomials to vanish, in particular Fn + i; 2) when one of the intermediate polynomials Fi, Ft, • • • , Vn vanishes, the number of changes of sign in the series (7) is not altered, lor if Vr vanishes for x = c, Vr-i and Fr + i have different signs for x = c. It follows that the number of changes of sign in the series (7) remains the same, eioept when z passes through a root of F = 0. If Fi/ F passes from + 00 to — oo, this number increases by one, but it diminishes by one on the other hand if Vi/V passes from - 00 to +00. Hence the index Is equal to the difference of the number of changes of sign in the series (7) for x = 6 and x = a. 78. Aret of a curve. We can now give a purely analytic definition of the area bounded by a continuous plane curve, the area of the notangle only being considered known. For this purpose we need IV. §78] ALLIED GEOMETRICAL CONCEPTS 159 only translate into geometrical language the results of $ 72. Let f(x) be a function which is continuous in the closed interval (a, 6), and let us suppose for definiteness that a<b and that fix) > in the interval. Let us consider, as above (Fig. 9, % 65), the portion of the plane bounded by the contour AMBBqA^^ composed of the seg- ment .-io/iu of the X axis, the straight lines AA^ and BB^ parallel to tlie y axis, and having the abscissa; a and 6, and the arc of the curve A MB whose ecjuation is y =f(x). Let us mark off on -.^o^o a certain number of points of division Pj, Pj, • • , Pi^i, Pa "t whose abscissa.' are a-,, Xj, • ••, a;^.,, a*,, •••, and through these points let us draw parallels to the y axis which meet the arc A MB in the points Qu Qtf •••» Qi-iy Qi) •••> respectively. Let us then consider, in particular, the portion of the plane bounded by the contour Qi-iQiPiPi~iQi-ii and let us mark upon the arc Q,_iQ.- the highest and the lowest points, that is, the points which correspond to the maximum 3/,- and to the minimum w,. of f(x) in the interval (ir,_,, Xf). (In the figure the lowest point coincides with Q,_i.) Let /ij be the area of the rectangle Pf_i /',*,«._ i erected upon the base Pi. I Pi with the altitude 3/„ and let r,- be the area of the rectangle Pi-iPiQiQi-i erected upon tlie base Pi^iPi with the alti- tude 7/1,. Then we have Ri = Mi(Xi - ar,_,), r^ = m,.(a;< - x,_,), and the results found above (§ 72) may now be stated as follows : whatever be the points of division, there exists a fixed number / which is always less than 2/^, and greater than 2r,, and the two sums 2/f,- and 2r, approach / as the number of sabintervals P<_i/*i increases in such a way that each of them approaches zero. We shall call this common limit / of the two sums 2/?, and 2r, the area of the portion of the plane hounded by the contour AMBBqA^A. Thus the area under consideration is defined to be equal to the definite integral j^f{x)dx. This definition agrees with the ordinary notion of the area of a plane curve. For one of the clearest points of this rather vague notion is that the area bounded by the contour P<_i/*<<2,n<Q<_,/*,_, lies Ixjtween the two areas /?< and r, of the two rectangles P,_iP,*,«,-.i and P,-i/*,7,Q,_i; hence the total area bounded by the contour AMBBqAqA must surely be a quantity which lies between the two sums 2/?, and 2r,. But the definite integral / is the •nly fixed quan- tity which always lies between these two sums for any mode of subdivision of A^B^y since it is the common limit of 2/^^ and 2r<. 160 DEFINITE INTEGRALS [IV, §79 The given area may also be defined in an infinite number of other ways as the limit of a sum of rectangles. Thus we have seen that the definite integral / is also the limit of the sum 2(^,-aj,_i)/(«, where ^, is any value whatever in the interval (ic,_i, x,). But the element (x.-^i-,)/(f) of this sum represents the area of a rectangle whose base is Pi^iPi and whose altitude is the ordinate of any point of the arc Qi.itiiQi. It should be noticed also that the definite integral / represents the area, whatever be the position of the arc A MB with respect to the X axis, provided that we adopt the convention made in § 67. Every definite integral therefore represents an area ; hence the calcu- lation of such an integral is called a quadrature. The notion of area thus having been made rigorous once for all, there remains no reason why it should not be used in certain arguments which it renders nearly intuitive. For instance, it is perfectly clear that the area considered above lies between the areas of the two rectangles which have the common base AqBq, and which have the least and the greatest of the ordinates of the arc A MB, respectively, as their altitudes. It is therefore equal to the area of a rectangle whose base is AqBq and whose altitude is the ordinate of a properly chosen point upon the arc A MB, — which is a restate- ment of the first law of the mean for integrals. 79. The following remark is also important. Let /(a;) be a func- tion which is finite in the interval (a, b) and which is discontinuous in the manner described below for a finite number of values between a and b. Let us suppose that f(x) is continuous from c to c-\-k(k>0), and that /(c -}- c) approaches a cer- tain limit, which we shall denote by f(c + 0), as e approaches zero through positive values; and like- wise let us suppose that f(x) is continuous between c — k and c and that f(c — e) approaches a limit /(« — 0) as c approaches zero through positive values. If the two limits f(r. + 0) and /(o - 0) are different, the function f(x) is dis- continuoufl for x = e. It is usually agreed to take for /(c) the IV, 5 80] ALLIED GEOMETRICAL CONCEPTS 161 value [f(c + 0) -\-f{e - 0)]/2. If the function /(z) has a certain number of points of discontinuity of this kind, it will be repre- sented graphically by several distinct arcs AC^ CD, D'B. Let o and d, for example, be the abscissa) of the points of discontinuity. Then we shall write f(x)dx = / f(x)dx -f / f{x)dx + f f{x)dx, «/• */«r Jd in accordance with the definitions of § 72. Geometrically, this taninie integral represents the area bounded by the contour A CC'IjD'HHqAqA. If the upper limit b now be replaced by the variable x, the definite integral n^)=jj<^ x^dx is still a continuous function of x. In a point x where f(x) is con- tinuous we still have F'(x) = f(x). For a point of discontinuity, X = c for example, we shall have Xe + h f(x) dx = hf(c + Bh), < d < 1, and the ratio [F(c -h h)— F(c)'\/h approaches /(c -f 0) or/(r — 0) according as h is positive or negative. This is an example of a function F(x) whose derivative has two distinct values for certain values of the variable. 80. Length of a curvilinear arc. Given a curvilinear arc AB; let us take a certain number of intermediate points on this arc, m^, w,, •'> ''^n-i » ^"d let us construct the broken line ylmjm, •• • m,_,B by connecting each pair of consecutive points by a straight line. If the length of the perimeter of this broken line approaches a limit as the number of sides increases in such a way that each of them approaches zero, this limit is defined to be the length of the arc AB. Let be the rectangular coordinates of a point of the .uv Ati expressed in terms of a parameter t, and let us suppose that as / varies from a to b{a<b) the functions /, ^, and ^ are continuous and possess continuous first derivatives, and that the point (x, y, x) describes the arc AB without changing the sense of its motion. Let 162 DEFINITE INTEGRALS [IV, §80 be the values of t which correspond to the vertices of the broken Una Then the side c, is given by the formula or, applying the law of the mean to «,• — aj^^i, • •, where ^,, 17^ C.- li© between ^,_i and ^f. When the interval {ti_^, i^) is very small the radical differs very little from the expression v[/'(«.-.)]' + [-^'(«.-i)P + [fc'.-or- In order to estimate the error we may write it in the form But we have and consequently /^fe)+/fe-0 <1. Hence, if each of the intervals be made so small that the oscillation of each of the functions f'(t), <f>'(t), \l/'(t) is less than e/3 in any interval, we shall have where h.i<«i and the perimeter of the broken line is therefore equal to The supplementary term 2€,(^< — ^^.i) is less in absolute value than €l(ti — ^<_,), that is, than €(* — a). Since € may be taken as small as we please, provided that the intervals be taken sufficiently •mall, it follows that this term approaches zero ; hence the length S of the arc i4B is equal to the definite integral (8) S= C V/'a + <^'^ + .A" dt. This definition may be extended to the case where the derivatives /, ^', ^' are discontinuous in a finite number of points of the arc AB, IV, §80] ALLIED GEOMETRICAL CONCEPTS 168 which occurs when the curve has one or more comers. We need only tlivide the arc AB into several parts for each of which/', ^', ^' are continuous. It results from the formula (8) that the length S of the arc between a fixed point A and a variable point M, which corresponds to a value t of the parameter, is a function of t whose derivative is whence, squaring and multiplying by eft*, we find the formula (9) dS^ = rfx'* 4- rfy* + dz^, which does not involve the independent variable. It is also easily remembered from its geometrical meaning, for it means that dS is the diagonal of a rectangular parallelopiped whose adjacent edges are dXf di/j dz. Note. Applying the first law of the mean for integrals to the definite integral which represents the arc MqMh whose extremities correspond to the values t^^ t^ of the parameter (tx > t^)^ we find * = arc M^I, = (t, - to) ^f\e) -f <t>'\e) 4- ^'\e), where lies in the interval (^„, ti). On the other hand, denoting the chord ^l^i^Il by c, we have Applying the law of the mean for derivatives to each of the differ- ences /(^i) — /(^»), • •, we obtain the formula c==(h- to) Vf'\$) -h ^'\f,) -f ^'\0y where the three numbers ^, 17, ( belong to the interval (t^t <i). By the above calculation the difference of the two radicals is less than c, provided that tlie oscillation of each of the functions/'(<), ^'(0» ^'(0 is less than </3 in the interval (^o* U)- Consequently we have »-c<c(<, - to), or, finally, If the arc ^fa^fl is infinitesimal, <t~ '0 approaches zero; hence c, iuid therefore also 1 — r/s, approaches aero. It follows that ths ratio of an injinitesinuil arc to its chord approaches unity as its limit. 164 DEFINITE INTEGRALS [IV, §81 ExampU. Let us find the length of an arc of a plane curve whose equation in polar coordinates is p =/(w). Taking w as independent variable, the curve is represented by the three equations a; = p cos w, y = p sin o», ;e = ; hence da* = dx^ -f rfy* =(cos o) rfp — p sin <u d<tif + (sin mdp -\- p cos w dmy, or, simplifying, ds^ = dp^'{-p''dio\ Let us consider, for instance, the cardioid, whose equation is p = R -\- Rcos w. By the preceding formula we have ds^ = R^dw^ [sin^o) + (1 + cos a>)2] = 4:R^ cos^ | cio)*, or, letting q> vary from to tt only, ds = 2R cos — c?<u ; and the length of the arc is (4..sin|)"\ where u>o and <oi are the polar angles which correspond to the extrem- ities of the arc. The total length of the curve is therefore 8 R. 81. Direction cosines. In studying the properties of a curve we are often led to take the arc itself as the independent variable. Let us choose a certain sense along the curve as positive, and denote by s the length of the arc AM between a certain fixed point A and a vari- able point 3/, the sign being taken + or — according as M lies in the positive or in the negative direction from A. At any point M of the curve let us take the direction of the tangent which coincides with the direction in which the arc is increasing, and let a, ^, y be the angles which this direction makes with the positive directions of tlie three rectangular axes Ox, Oy, Oz. Then we shall have the following relations : cos or _ cos /8 cosy _ 1 ±1 dx " dy ^ dz ^ ^dx'' -f dy^ -f"^ ^ ~di' To find which sign to take, suppose that the positive direction of the tangent makes an acute angle with the x axis ; then x and s inorea«e simultaneously, and the sign -f should be taken. If the angle a is obtuse, cos a is negative, x decreases as s increases, dx/ds IV, $82] ALLIED GEOMETRICAL CONCEFFS IfiS is negative, and the sign -f should be taken again, iience in any case the following formula; hold : (10) C08a=— , COS^ = -^, ^y^^, where rfx, rfy, rf«, ds are differentials taken with respect to the same independent variable, which is otherwise arbitrary. 82. Variation of a segment of a straight line. Let AfM^ be a segment of a straight line whose extremities describe two curves C, f,. On each of the two curves let us choose a point as origin and a positive sense of motion, and let us adopt the follow- ing notation : «, the arc AM \ «!, the arc AiMxi — the two arcs being taken with the same sign ; /, the length MM^ ; ^, the angle between MM^ and the positive di- rection of the tangent 3/7'; 6^, the angle between Mi M and the positive direction of the tangent M^ 7\. We proceed to try to find a relation between B, $i and the differentials ds, rf#,, dl. Let (x, y, «), (jcx, yi, z^) be the coordinates of the points M, Mi, respectively, a, ft, y the direction angles of M7\ and a,, fii, yj the direction angles of Mi 7\. Then we have P = (x-xiy-^(y-yiy-^(z-^ziy, from which we may derive the formula Idl = (x -Xi)(dx - dxi) + (y - yx)(dy - dyi) 4- (« - «,)(<& - <i«,), which, by means of the formulae (10) and the analogous formulas for 6\, may be written in the form dl = (^^C03aH-^^COS/9 + ^^C08y)<i« -f (^^ cos a, + ^^ cos ft + ^S^ cos y,) rf*,. But (x — Xi)/l, {y — y\)/li (« — «i)/^ are the direction cosines of Ml M, and consequently the coefficient of <i5 is ~ cos 6. Likewise the coefficient of (^i is — cos ^; hence the desired relation is (10') dl = - ds cos $ - dsioos $1. We shall make frequent applications of this formula; one such we proceed to discuss immediately. 166 DEFINITE INTEGRALS [IV, §88 8S. Theorems of Graves and of Chasles. Let E and E' be two confocal ellipses, aod let the two tangents MA, MB to the interior ellipse E be drawn from a point 3f, which lies on the exterior ellipse E\ The difference MA + MB — arc ANB remains con- stant as the point M describes the ellipse E\ Let s and s' denote the arcs OA and OB, a the arc (XM, I and V the distances AM and BM, 6 the angle between MB and the positive direction of the tangent MT. Since the ellipses are confocal the angle between MA and MT is p. -„ equal to it — 0. Noting that AM coincides with the positive direction of the tangent at A, and that BM is the negative direction of the tangent at B, we find from the formula (10'), successively, dl = — ds ■\- dff cos 6 , dl' = ds' — d<r cos ; whence, adding, d{l+V)=d (8' -s)=d (arc ANB), which proves the proposition stated above. The above theorem is due to an English geometrician, Graves. The following theorem, discovered by Chasles, may be proved in a similar manner. Given an ellipse and a confocal hyperbola which meets it at N. If from a point M on that branch of the hyi)erbola which passes through N the two tangents MA and MB be drawn to the ellipse, the difference of the arcs NA — NB will be equal to the difference of the tangents MA — MB. m. CHANGE OF VARIABLE INTEGRATION BY PARTS A large number of definite integrals which cannot be evaluated directly yield to the two general processes which we shall discuss in this section. 84- Change of variable. If in the definite integral ///(«) dx the variable x be replaced by a new independent variable t by means of the substitution x = <l>(t), a new definite integral is obtained. Let U8 suppose that the function <^(^) is continuous and possesses a continuous derivative between a and jS, and that <^(^) proceeds from a to 6 without changing sense as t goes from a to p. The interval (a, P) having been broken up into subintervals by the intermediate values a, t^, t^, . . ., ^„_,, ^, let a, x^,x^, - • ., £c„_i, b be the corresponding values of a; = <^(^). Then, by the law of the mean, we shall have where $^ lies between f,_, and tf. Let ft •-= <^(d,.) be the corresponding falue of « which lies between x^^i and a,. Then the sum IV,}*»J CHANGE OF V A 11 TABLE 167 approaches the given definite integral as its limit But this sum may also be written and in this form we see that it approaches the new definite integral X /[*(<)]♦'(<)<« as its limit. This establishes the equality (U) fAx)dx = /"/[*(<)] *'(0*. •/a »/« which is called the foT*mula for the change of variable. It is to be observed that the new differential under the sign of integration is obtained by replacing x and dx in the differential /(x)rfx by their values <^(^) and <f>'{t)dt^ while the new limits of integration are the values of t which correspond to the old limits. By a suitable choice of the function <^(^) the new integral may turn out to be easier to evaluate than the old, but it is impossible to lay down any definite rules in the matter. Let us take tlie definite integral i (x-ay-i-p^' for instance, and let us make the substitution x = a -{- fit. It becomes p or, returning to the variable x, 1/ x — a ^ ^ a\ 7, arc tan — 1- arc tan - 1. ^\ P P/ Not all the hypotheses made in establishing the formula (11) were necessary. Thus it is not necessary that the function <f>(t) should always moye in the same sense as t varies from a to fi. For defi- niteness let us suppose that as t increases from <t to y (y < j8), <f>(t) steadily increases from a to c (c>b)\ then as t increases from y to fi, <f>(t) decreases from c to b. If the function /(«) is continuous in the interval (a, r), the formula may be applied to each of the inter- vals (a, c), (r*, l})y whicli gives 168 DEFINITE INTEGRALS tIV,§84 or, adding, fjXx)dx = ( f[.'t>{t)-]^Xt)dt. On the other hand, it is quite necessary that the function </>(^) should be uniquely defined for all values of t. If this condition be disregarded, fallacies may arise. For instance, if the formula be applied to the integral j^^ dxj using the transformation x = ^'^, we should be led to write //^=/ \-Jtdt, which is evidently incorrect, since the second integral vanishes. In order to apply the formula correctly we must divide the interval (— 1, -f 1) into the two intervals (— 1, 0), (0, 1). In the first of these we should take a; = — V? and let t vary from 1 to 0. In the second half interval we should take x = V^ and let t vary from to 1. We then find a correct result, namely NoU, If the upper limits h and ^ be replaced by x and t in the formula (11), it becomes f'jlx)dx=f/ii,(t)2,l,Xt)dt, %Ja %J a which shows that the transformation x = <f>(t) carries a function F(ar), whose derivative is /(x), into a function ^(t) whose derivative "/[^(O] ^'(0* "^^^^ ^^o follows at once from the formula for the derivative of a function of a function. Hence we may write, in general, jA^)dx^jfi4^{t)-]4>\t)dt, which is the formula for the change of variable in indefinite integrals. IV, iM] INTEGRATION BY PARTS 169 85. Integration by parts. Let u and v be two functions which, togetlier witli tlieir derivatives u' and v', are continuous between a and b. Then we have d(uv) dv . du — ^ — ^ = u — -f- V — » dx dx dx whence, integrating both sides of this equation, we find Ja dx J^ dx J^ dx This may be written in the form (12) / udv = \uv']\- I vdUy %Ja %Ja where the symbol \F{^x)\ denotes, in general, the difference F(i)-F(a). If we replace the limit 6 by a variable limit x, but keep the limit a constant, wliich amounts to passing from definite to indefinite inte- grals, this formula becomes (13) I udv = uv — I t' du. Thus the calculation of the integral J udv is reduced to the cal- culation of the integral fvdu, which may be easier. Let us try, for example, to calculate the definite integral I x^logxcfoj, w -f 1 ^ 0. Setting u = logx, v = x'"-^^ /(tn -f 1), the formula (12) gives -[ g"-^'loga; _ af^' T m-fl (w-fl)*J.' This formula is not applicable if wt -f- 1 = ; in that particular case we have /'.og 4^ =[.; (log ,).];. It is possible to generalize the formula (12). Let the succes- sive derivatives of the two functions u and v be represented by u', u", . ., tt<"+»>; v', w", ••, r<" + »>. Then the application of the 170 DEFINITE INTEGRALS [IV, §85 formula (12) to the integrals /Mrft/"^ /w'<^v^'*"^^ ••• leads to the following equations: X6 •»& />& %Ja *Ja dx. Multiplying these equations through by +1 and —1 alternately, and then adding, we find the formula (14) + (-l)'' + i I u^^'^^^vdxy which reduces the calculation of the integral Juv^'^'^^^dx to the cal- culation of the integral fu^^-^^^vdx. In particular this formula applies when the function under the integral sign is the product of a polynomial of at most the nth degree and the derivative of order (n + 1) of a known function v. For then m<'' + '> = 0, and the second member contains no integral signs. Suppose, for instance, that we wished to evaluate the definite integral £e^'f{x)dx, where /(x) is a polynomial of degree n. Setting w =f(x), v = e'*'Y<»>""^S the formula (14) takes the following form after ef^"" has been taken out as a factor : The same method, or, what amounts to the same thing, a series of integrations by parts, enables us to evaluate the definite integrals J co%mxf(x)dXy j smmxf(x)dx, where /(«) is a polynomial. IV, J 8(5] INTEGRATION BY PARTS 171 86. Taylor's series with a remainder. In the formula (14) let us replace u by a function F(x) which, together with its first n + 1 derivatives, is continuous between a and 6, and let us set v = (b—xy. Then we have v' = - n{b - xy-\ v" = n(n - l)(b - x)--«, • • ., v<-)=(~l)-1.2...n, t;<--^») = 0, and, noticing that r, v', v", •., «;<■-*) vanish for x=b,we obtain the following equation from the general formula : =(- l)"! n!F(^») - n\F(a) - n\F'(a) (b - a) - ~ F"(a)(6 - ay---- F^^\a)(b - a)-l -h (- 1)- -^ ' r f <- ^ '\x) (b - xydxy which leads to the equation Since the factor (6 — x)" keeps the same sign as x varies from a to i, we may apply the law of the mean to the integral on the right, which gives F"*'\x){b - xydx = f *"+"({) / (* - xydx where ^ lies between a and b. Substituting this value in the preced- ing equation, we find again exactly Taylor's formula, with Lagrange's form of the remainder. 87. TraDScendenUl character of e. From the formula (15) we can prove a famouK theorem due to llermite : The number e is not a root qf any algebraic equation whose co^cienta are all intcgertt.* Setting a = and w = - 1 in the formula (16), it becomes J^«-«/(«)ci»=-C«-«jr(x)]J, * The present proof ii doe to D. Hilbert. who drew hir inspiration from the method used by Hermite. 172 DEFINITE INTEGRALS [IV, §87 idMM F(x) =f(x) +r{x) + '" +/<''>(aj); and ibiB again may be written in the form (16) F(b) = «*F(0) - ^f/{^) e-'^dx. Now let us suppoee that e were the root of an algebraic equation whose coeflB- cienu are all integers : Co + Cie + cae2 + . . . + c^e™ = 0. TbeUf setting 6 = 0, 1, 2, • • • , m, successively, in the formula (16), and adding the results obtained, after multiplying them respectively by Co, Ci, • • -, c^, we obtain the equation (17) coF(0) + c, ^(1) + . . • + c„ F{m) + ^ CieQ{x)e-^dx = 0, »=o where the index t takes on only the integral values 0, 1, 2, • • • , m. We proceed to show that such a relation is impossible if the polynomial /(x), which is up to the present arbitrary, be properly chosen. I«et us choose it as follows : •^(*) = , ^ ,., ^^~Hx - 1)^(« -2)P"-{x- m)p, (p-l)l where p is a prime number greater than m. This polynomial is of degree mp + p — 1, and all of the coefficients of its successive derivatives past the pth are integral multiples of p, since the product of p successive integers is divisible by p!. Moreover /(x), together with its first (p — 1) derivatives, vanishes for X = 1, 2, . • • , m, and it follows that F{1), F{2), • • • , F(m) are all integral mul- tiples of p. It only remains to calculate .F(O), that is, F(0) =/(0) +/'(0) + . . . +f(p-^^0) +/(i')(0) +/(P + i)(0) + . . .. In the first place, /(O) = /'(O) = . . • =/(p-2)(0) = 0, while /(p)(0), /^^ + i)(0), • • • are all integral multiples of p, as we have just shown. To find /(p - 1) (0) we need only multiply the coefficient of xp-i in/(x) by (p - 1) !, which gives ± (1 . 2 • • • m)P. Hence the sum CoF(0) + CiF(l) + --- + c«F(m) b equal to an Integral multiple of p increased by ± Co(l . 2 . • . m)p. If p be taken greater than either m or Co, the above number cannot be divisible by p ; hence the first portion of the sum (17) will be an integer different from zero. We shall now thow that the sum Vc.c^ r/(x)e-'(te eao be made ■mailer than any preaaeigned quantity by taking p sufficiently large. Am z Tariee from to i each factor of /(x) is less than in ; hence we have IV,}*!!] INTEOUATION BY PARTS 178 '^*'"<(F^rij] "'■'*'-'• '+#-1. xc^j;/(x) IJo''^' r(p-l)! Jo ^(p- 1)1 from which it follows that e-'dx <jr^__-.e- = 0(p), where If is an upper limit of |co | + | Ci | + h | c. | • Ae p increaoes indefi- nitely the function <f> (p) approaches zero, for it is the general term of a conrer- gent series in which the ratio of one term to the preceding approaches zero. It follows that we can find a prime number p so large that the equation (17) is impossible ; hence Hermite's theorem is proved. 8S. Legendre's polynomials. Let us consider the integral /. QP«dx, where P^ (z) is a polynomial of degree n and Q is a polynomial of degree leei than n, and let us try to determine P«,(x) in such a way that the integral van- ishes for any polynomial Q. We may consider P„ (z) as the nth derivative of a polynomial R of degree 2»i, and this polynomial li is not completely determined, for we may add to it an arbitrary polynomial of degree (n - 1) without changing its nth derivative. We may therefore set 2^= d^'R/dz"^ where the polynomial /?, together with its first (n — 1) derivatives, vanishes for z = a. But integrating by parts we find Ja dz* L d«-» ^ dx-« ^ d*--U. and since, by hypothesis, «(a) = 0, /?'(a) = 0, , «<--»(a)=0, the expression 'Q(6)f^«-')(6) - Q'(6)/J<'.-t>(6)-f •• ± Q(— ')(6)iJ(6) must also vanish if the integral is to vanish. Since the polynomial q of degree n - 1 Is to be arbitrary, the quantitiea Q(^)i Q'i^)* •• •> 0^""'^^) ^^ themselves arbitrary; hence we must also have R (6) = 0, R\b) =0, .... iJ<« - »)(6) = 0. The polynomial R (z) is therefore equal, save for a conctant factor, to the product (z — o)*(z - 6)" ; and the required polynomial Pmi!'') is completely determined, •ave for a constant factor, in the form If the limits a and 6 are — 1 and + 1, respectively, the polynomials P« are Legendre^s polynomials. Choosing the constant C with Legendre, we will set 174 DEFINITE INTEGRALS [IV, §88 If we alao agree to set Xo = 1, we shall have ^ 3xa-l ^ 6x3 -3x jro = i, Jri = x, Xt = — - — 1 ^8 = — - — ' In general, X, ta a polynomial of degree n, all the exponents of x being even or odd with n. Leibniz' formula for the nth derivative of a product of tv^o factors (S 17) gi^M ^ 0°^ ^® formula (W) -r,(l) = 1, X.(- 1) = (- 1)». By the general property established above, (90) J^ Xn<f>{x)dx = 0, where i>{z) is any polynomial of degree less than n. In particular, if m and n are two different integers, we shall always have (21) f^XmXndx^O. This formula enables us to establish a very simple recurrent formula between three successive polynomials X„. Observing that any polynomial of degree n can be written as a linear function of Xo, Xi, • • . , X„, it is clear that we may set XX, = CoXn + l + CiXn + CiXn-l + 0^X^-2 +"', where Co, Ci, Ca, • • • are constants. In order to find Cg, for example, let us multiply both sides of this equation by Xn-2, and then integrate between the limita — 1 and + 1. By virtue of (20) and (21), all that remains is Cs£'[x;_2dx = 0, and hence Cz = 0. It may be shown in the same manner that C4 = 0, Cs = 0, • • . The coefficient Ci is zero also, since the product xX„ does not contain x». Finally, to find Co and Ca we need only equate the coefficients of x" + ^ and then equate the two sides f or x = L Doing this, we obtain the recurrent formula (S2) (n + l)X,+i - (2n + l)xXn + nX„_i = 0, which affords a simple means of calculating the polynomials X„ successively. The reUtion (22) shows that the sequence of polynomials (48) Xo, Xi, Xa, •••, Xn the properties of a Sturm sequence. As x varies continuously from - 1 to 4- 1| the number of changes of sign in this sequence is unaltered except when through a root of X„ = 0. But the formula (19) show that there are n of sign in the sequence (23) for x = - 1, and none for x = 1. Hence the eqaatloD X. = has n real roots between - 1 and + 1, which also readily follows from Rollers theorem. IV, §89] IMPROPER AND LIKE INTEGRALS 176 IV. GENERALIZATIONS OF THE IDEA OF AN INTEGRAL IMPROPER INTEGRALS LINE INTEGRALS • 89. The integrand becomes infinite. Up to the present we have sup- posed that the integrand remained finite between the limits of inte- gration. In certain cases, however, the definition may be extended to functions which become infinite l>etween the limits. Let us first consider the following particular case : f{x) is continuous for every value of X which lies between a and 6, and for x = 6, but it becomes infinite for x — a. We will suppose for definiteness that a < 6. Then the integral of /(a;) taken between the limits a -|- c and h (<>0) has a definite value, no matter how small c be taken. If this integral approaches a limit as < approaches zero, it is usual and natural to denote that limit by the symbol X \x)dx. If a primitive ai f{x)y say ^(-c), be known, we may write f and it is sufficient to examine F(a -\- e) for convergence toward a limit as e approaches zero. We have, for example, f Mdx _ M r 1 !_■] . If fi > 1, the term l/c**"* increases indefinitely as e approaches zero. But if ^ is less than unity, we may write l/c^-*= e'"**, and it is clear that -this term approaches zero with e. Hence in this case the definite integral approaches a limit, and we may write ' Mdx ^ 3/(&~aV-^ {x — ay 1 - ^ If ^ — 1, \v.> liave X' and the right-hand side increases indefinitely when c approaches zero. To sum up, the necessary and stiffirient condition that the given int&' gral should approach a limit is that fi should be less than unity. * It is poMible, if desired, to read the next diapter before reading the cloeiDg sw^ tioDS of this chapter. 176 DEFINITE INTEGRALS [IV, §89 The straight line ar = a is an asymptote of the curve whose equa- tion is M if /A is positive. It follows from the above that the area bounded by the X axis, the fixed line x = i, the curve, and its asymptote, has a finite value provided that /x < 1. If a primitive of f(x) is not known, we may compare the given integral with known integrals. The above integral is usually taken as a comparison integral, which leads to certain practical rules which are sufficient in many cases. In the first place, the upper limit b does not enter into the reasoning, since everything depends upon the manner in which f{x) becomes infinite for x~ a. We may therefore replace h by any number whatever between a and h, which amounts to writing /, , = X*^ , + X * ■'•^ particular, unless f{x) has an infi- nite number of roots near ic = a, we may suppose that f{x) keeps the same sign between a and c. We will first prove the following lemma ; Let <^(x) be a function which is positive in the interval (a, b), and suppose that the integral J^ ^<t>(x) dx approaches a limit as c approaches zero. Then, if \f(x) | < <^(cc) throughout the whole inter- valf the definite integral f^f(x)dx also approaches a limit. lff(x) is positive throughout the interval (a, b), the demonstration is injimediate. For, since f(x) is less than </> («), we have f(x)dx < I <f>(x)dx. Moreover J^^^f(x)dx increases as e diminishes, since all of its ele- ments are positive. But the above inequality shows that it is con- stantly less than the second integral ; hence it also approaches a limit If f(x) were always negative between a and b, it would be necessary merely to change the sign of each element. Finally, if the function f(x) has an infinite number of roots near x = a, we may write down the equation £^A^)dx ^j^^ifix) + I f(x) \]da^ -J' \f(x) I dx. The Moond integral on the right approaches a limit, since !/(«)! <^(«). Now the function /(ic) + |/(x)| is either positive IV. §89] IMPROPKR AND LINE INTEGRALS 177 or zero between a and b, and its value cannot exceed 2 ^(2); hence the integral f i/w+i/(»')i]«fa «/a-f« also approaches a limit, and the lemma is proved. It follows from the above that if a function /(x) does not approach any limit whatever for x = a, but always remains less than a fixed number, the integral approaches a limit. Thus the inte^^ral ^* sin (1/x) dx has a perfectly definite value. Practical rule. Suppose that the function f(x) can be written in the form f(x)= ^(^) , -^"^ ^ (x-a)'' where the function ^(a;) remains finite when x approaches a. If fjL<l and the function ^(x) remains less in absolute value than a fixed number A/, the integral approaches a limit. But if fi>l and the absolute value of \f/(x) is greater than a positive number w, the integral approac/ies 710 limit. The first part of the theorem is very easy to prove, for the abso- lute value of f{x) is less than Af/(x — a)**, and the integral of the latter function approaches a limit, since fi<l. In order to prove the second part, let us first observe that ^(ar) keeps the same sign near x = a^ since its absolute value always exceeds a positive number m. We shall suppose that ^(x)>0 between a and b. Then we may write £/<">- >x;.F^ and the second integral increases indefinitely as c decreases. These rules are sufficient for all cases in which we can find an exponent fi such that the product (x — aYf{x) approa^'hes, for X = a, a limit A' different from zero. If fi is less than unity, the limit b may be taken so near a that the inequality holds inside the interval (a, 6), where Z. is a positive number graatar 178 DEFINITE INTEGRALS [IV, §89 than I A" |. Hence the integral approaches a limit On the other hand, if ^ > 1, 6 may be taken so near to a that inside the interval (a, h), where I is a positive number less than |iir|. Moreover the function f{x), being continuous, keeps the same sign ; hence the integral /^"^^ /(a:) c^x increases indefinitely in absolute value.* Examples. Let /(a;) = P/Q be a rational function. If a is a root of order m of the denominator, the product {x - aYf{x) approaches a limit different from zero for x = a. Since m is at least equal to unity, it is clear that the integral £^^^/(x)<;a; increases beyond all limit as c approaches zero. But if we consider the function where P and R are two polynomials and R{x) is prime to its deriv- ative, the product (x — ay^^f(x) approaches a limit for a; = a if a is a root of R(x), and the integral itself approaches a limit. Thus the integral dx f. i + < vn^ approaches 7r/2 as c approaches zero. Again, consider the integral J^^logxdx. The product a;^/^loga; has the limit zero. Starting with a sufficiently small value of x, we may therefore write log x < Mx' ^/^, where 3/ is a positive number chosen at random. Hence the integral approaches a limit. Everything which has been stated for the lower limit a may be repeated without modification for the upper limit b. If the function / (x) is infinite for a; = 6, we would define the integral j^ /(«) dx to be the limit of the mtegvdX j^~' f(x)dx as «' approaches zero. lif(x) is infinite at each limit, we would define J VC^)^^ ^is the limit of the integral^ ~*f(x)dx as c and t' both approach zero independ- ently of each other. Let c be any number between a and b. Then we may write *The flrat part of the proposition may also be stated as follows: the integral has a limit If an exponent m can \w fiuind (0 < m < 1) such that the product (x — a)'*/(x) approaclMt a limit ^ aa x u^tpruaches a, — the case where A = uot being excluded. IV,}90] IMPROPER AND LINE INTEGRALS 179 r"f{x)dx = r f(x)dx + r"f(?:)dx, and each of the integrals on the right should approach a limit in this case. Finally, if f(x) becomes infinite for a value e between a and b, we would define the integral j^f{x)dx as the sum of the limits of the two integrals j^~*f{x)dx, j^ ^f{x)dxy and we would proceed in a similar manner if any number of discontinuities whatever lay between a and b. It should be noted that the fundamental formula (6), which was established under the assumption that f{x) was continuous between a and h^ still holds when f(x) becomes infinite between these limite, provided that the primitive function F(x) remains continuous. For the sake of definiteness let us suppose that the function /(ar) becomes infinite for just one value c between a and b. Then we have f(x) dx = \im I f{x) (& + lim / f{x) dx ; and if F(x) is a primitive of /(x), this may be written as follows : Xf{x)dx = lim F(c -t!)- F(a) -f F{b) - lim F{c -|- c). Since the function F(x) is supposed continuous for ar = r, F(c + e) and F(c — «') have the same limit F(c), and the formula again becomes f(x)dx = F(b)-F(,a). £ The following example is illustrative : i £=[:-']:;-. If the primitive function F(x) itself becomes infinite l^etween a and b, the formula ceases to hold, for the integral on the left has as yet no meaning in that case. The formulae for change of variable and for integration by parts may be extended to the new kinds of integrals in a similar manner by considering them as the limits of ordinary integrals. 90. Infinite limits of integration. Let/(a;) be a function of x which is continuous for all values of x greater than a certain number a. Then the integral /'/(x)<ir, where / > a, has a definite value, no X80 DEFINITE INTEGRALS [IV, §90 matter how large I be taken. If this integral approaches a limit as / increases indefinitely, that limit is represented by the symbol X f(x)dx. If a primitive of f{x) be known, it is easy to decide whether the integral approaches a limit. For instance, in the example ^ dx i „ = arc tan I ^ + ^' the right-hand side approaches 7r/2 as I increases indefinitely, and this is expressed by writing the equation dx IT X l-{-x^ 2 Likewise, if a is positive and /x — 1 is different from zero, we have r'kdx^ k /J 1_\ If fi is greater than unity, the right-hand side approaches a limit as I increases indefinitely, and we may write r^'^ kdx ^ Ja X'^ k On the other hand, if fi is less than one, the integral increases indefi- nitely with l. The same is true for /x = 1, for the integral then results in a logarithm. When no primitive of f(x) is known, we again proceed by com- parison, noting that the lower limit a may be taken as large as we please. Our work will be based upon the following lemma : Let ^(x) be a function which is positive for x'> a, and suppose that the integral JJ <f> (x) dx approaches a limit. Then the integral J ^f(x) dx also approaches a limit provided that \f(x) \<<f»(x) for all values of X greater than a. The proof of this proposition is exactly similar to that given above. If the function f(x) can be put into the form wli«re the function ^{x) remains finite when x is infinite, the follow- ing theorems can be demonstrated, but we shall merely state them IV. §91] IMPROPER AND LINE INTEGRALS 181 If the absolute ralue o/\f/(^x) I'.s /'.s.s f/f/n >i fix*''l numhtr M ,i ml fi is (jreater than unity, the integral njipronrht^ a limit. If the absolute value of ^ (x) is greater than a potitive number m arul /i is less than or equal to unity^ the integral approaehee no limit. For instance, the integral X COB ax , ax /o 1+x^ approaches a limit, for the integrand may be written cos ax __ \ cos ax and the coefficient of 1/x* is less than unity in absolute value. The above rule is sufficient whenever we can find a positive num- ber /i for which the product x^f(x) approaches a limit different from zero as x becomes infinite. The integral approaches a limit if /i is greater than unity, but it approaches no limit if fi is less than or equal to unity.* For example, the necessary and sufficient condition that the inte- gral of a rational fraction approach a limit when the upper limit increases indefinitely is that the degree of the denominator should exceed that of the numerator by at least two units. Finally, if we take ' V/e(x) where P and R are two polynomials of degree p and r, respectively, the product a;'"/*~»'/(x) approaches a limit different from zero when X becomes infinite. The necessary and sufficient condition that the integral approach a limit is that p be less than r/2 —1. 91. The rules stated above are not aiw«iy8 sufiii'ient for deiermin- ing whether or not an integral approaches a limit. In the example /(j-) = (sin x)/x, for instance, tlie product x^f(x) approaches zero if fi is less than one, and can take on values greater than any given number if /i is greater than one. If /a = 1, it oscillates between + 1 and — 1. None of the above rules apply, but the integral does ap- proach a limit Let us consider the slightly more general integral * The integral also approaches a limit if the product a^/(x) (where m> 1) approadiag zero ai) x becomes intiuite. 182 DEFINITE INTEGRALS [IV, §91 -i: ■" dXf a>0. The integrand changes sign for x = kw. We are therefore led to study the alternating series (24) ao-a, + a^-a, + '-' + (-iya, + "', where the notation used is the following : ^^ sm X . -«^ dx J- /^Cn + l)* nv Substituting y + wtt for x, the general term a„ may be written »"=X'^" It is evident that the integrand decreases as n increases, and hence «•+!<"••• Moreover the general term a„ is less than J^ (l/7i7r)c?y, that is, than \/n. Hence the above series is convergent, since the absolute values of the terms decrease as we proceed in the series, and the general term approaches zero. If the upper limit / lies between nir and (n + 1) 77 , we shall have X' X where S^ denotes the sum of the first n terms of the series (24). As / increases indefinitely, n does the same, a„ approaches zero, and the integral approaches the sum S of the series (24). In a similar manner it may be shown that the integrals I sXnx^dx, I QO^x^dXy Jo which occur in the theory of diffraction, each have finite values. The curve y = sin x^, for example, has the undulating form of a sine curve, but the undula tions becom e sharper and sharper as we go out» since the difference v(n-i-l)7r — y/nir of two consecutive roots of sin x^ approaches zero as n increases indefinitely. Bimarlr. Thli last example gives rlae to an interesting remark. As x increases todsflBltoly sin x« oscillates between - 1 and +1. Hence an integral may approach a limit uvea if tlm integrand does not approach zero, that is, even if IV, i'-'J IMPUorKK AND LIXE INTEGRALS 188 the X axis Is not an a«y rnptote to tlie curve y — f{x). The following Is an example of the same kind in which the function /(z) does not change sign. The function m 1 + «• sin«x remains positive when x is positive, and it does not approach zero, since f{kn) - kit. In order to show that the integral approaches a limit, let us con- sider, as above, the series oo + ai + ••• + a, + --, where -L xax + x«ain«x As X varies from twr to (n + 1) «•, x« is constantly greater than n««*, and we may write a^<(n-|-l)«' \ %Jhw (-^»' dx 1 + n«»r«8ln2x A primitive function of the new integrand is arctan(Vl4- n»7r< tanx), and as x varies from nn to (n + 1) >r, tanx becomes infinite just once, p— nng from + 00 to — 00. Hence the new integral is equal (§ 77) to ir/Vl + n^««, and we have ^^ (n + l)>r« ^ (n-fl) It follows that the series Zom is convergent, and hence the integral fj/{x)dz approaches a limit. On the other hand, it is evident that the integral cannot approach any limit if /(x) approaches a limit h different from zero when x becomes infinite. For beyond a certain value of x, /(x) will be greater than | h/2 \ in absolute value and will not change sign. The preceding developments bear a close analogy to the treatment of infinite series. The intimate connection which exists between these two theories is brouj^ht out by a theorem of Cauchy's which will be considered later (Chapter VIII). We shall then also find new criteria which will enable us to determine whether or not an integral approaches a limit in more general cases than those treated above. 98. The function r(s). The definite integral . (26) r(a)= r^*x«->c-»dx has a determinate value provided that a is positive. Fur, let us consider the two integmls r x--»e-'<ix, rx«->e-«cfc. 184 DEFIXTTK INTEGRALS [IV, §93 when c is a Tery small positive number and I is a very large positive number. The aeoond integral always approaches a limit, for past a sufficiently large value of X we have x^-U-' < l/x*, that is, e^>x^-^K As for the first integral, the product x> - "/(3K) approaches the limit 1 as x approaches zero, and the necessary and sufficient condition that the integral approach a limit is that 1 - a be less unity, that is, that a be positive. Let us suppose this condition satisfied. the sum of these two limits is the function T{a), which is also called Euler^s inteffral of the second kind. This function r(a) becomes infinite as a approaches »ero, it is positive when a is positive, and it becomes infinite with a. It has a minimum for o = 1.4616321 •• •, and the corresponding value of T{a) is 0.8866082. •. Let us suppose that a> 1, and integrate by parts, considering e-*dx as the differential of - e-^. This gives r(o) = -[x«-ie-']^* + (a-l)J *a;«-2e-^(te, hut the product x^-^e-' vanishes at both limits, since a > 1, and there remains only the formula (26) r(a) = (a-l)r(a-l). The repeated application of this formula reduces the calculation of r(a) to the case in which the argument a lies between and 1. Moreover it is easy to determine the value of T{a) when a is an integer. For, in the first place, r(l) = j^''"e-dx = -[e-],+ * =1, and the foregoing formula therefore gives, for a = 2, 3, • • , n • • • , r(2) = r(i) = 1, r(3) = 2r(2) = i . 2 ; and, in general, if n is a positive integer, (27) r(n) = 1.2.3...(n-l) = (n-l)!. 93. Line integrals. Let ^J5 be an arc of a continuous plane curve, and let P(x, y) be a continuous function of the two variables x and y along AB, where x and y denote the coordinates of a point of AB with respect to a set of axes in its plane. On the arc AB let us take a certain number of points of division Wi, Wa, • • •, w-,, • • •, whose ootedinates are (a^x, y^), (ajj, y,), • • -, (a;,, y^), • • -, and then upon each of the arcs m^^jm^ let us choose another point n,- (^,, i;,) at random. Finally, let us consider the sum (28) S ^^^'' "^'^ (^» ~ *) + ^(^» ' ''«) (^« -^i)+" extended oyer all these partial intervals. When the number of points of diTision is increased indefinitely in such a way that each of the diiferenoes «< — x, _ , approaclies zero, the above sum approaches a IMPROPER AVn LIXK IXTEORALS 185 iuiiil which i.s oalltMl tl arc^ Ali, and which is r. u ut J\Xf y) extemied over the the Bymbol / Jab P{x, y)dx. In order to establish the existence of this limit, let us first sup- pose tliat a lino parallel to the y axis cannot meet the arc AH xn more tlian one point. Let a and b be the abscissae of the points A and Hy respectively, and let y = ^(aj) be the equation of Uie curve ^4 /J. Then ^(j*) is a continuous function of x in the interval (a, 6), by hypothesis, and if we replace y by <^(j-) in the function /'(ar, y), the resulting function ♦(x) = P\xj ^(^)] Js also continuous. Hence we have and the preceding sum may therefore be written in the form ♦(^i) (a-i - a) +*(f,) (x, - xO -f •• • + <I>(«(ar. - x,.,) + .... It follows that this sum approaches as its limit the ordinary definite integral %J a %J a a!i<] \v»' li !v.' filially the rnrmnl.i f I'(x,i,)Ux=f Plx,4,(x)-idx. Jab Ja If a line parallel to the y axis can meet the arc AB in more than one point, Sve should divide the arc into several portions, each of which is met in but one point by any line parallel to the y axis. If the given arc is of the form ACDIS (Fig. 14), for instance, where C and D are points at which the abscissa has an extremum, each of the arcs ACj CDy I>B satisfies the above condition, and we may write r P(x, y)dx = r P(x, y)dx + r P(x, y)djr-^ f P(x, y)dx, Javhb Jac Jcd Jdb But it should be noticed that in the calculation of the three integrals y ^D a !\ C A^ i I ! X Fio. 14 Ig5 DEFINITE INTEGRALS [IV, §93 on the right-hand side the variable y in the function P{x, y) must be replaced by three different functions of the variable Xy respectively. Curvilinear integrals of the form J^^ Q,(x, y) dy may be defined in a similar manner. It is clear that these integrals reduce at once to ordinary definite integrals, but their usefulness justifies their introduction. We may also remark that the arc AB may be com- posed of portions of different curves, such as straight lines, arcs of circles, and so on. A case which occurs frequently in practice is that in which the coordinates of a point of the curve AB are given as functions of a variable parameter where <^(^) and \l/(t), together with their derivatives <^'(^) and \f/\t), are continuous functions of t. We shall suppose that as t varies from a to p the point (x, y) describes the arc AB without changing the sense of its motion. Let the interval (a, p) be divided into a certain nimiber of subintervals, and let ^,-_ i and t^ be two consecu- tive values of ^ to which correspond, upon the arc AB, two points m,._, and m,- whose coordinates are (ic,_i, 2/,_i) and (a,-, y^), respec- tively. Then we have where d,- lies between ^,._i and ^,. To this value Oi there corresponds a point (^,-, i;,) of the arc m,_im,; hence we may write SP(^o 1,,) (a:, - x,_,) = SP[<^(eO. «AW)] <^'(^i) (ti - *i-i)> or, passing to the limit, / Pix, y)dx= f P[<f>(t), ,/r(0] <t>\t)dt. Jab Ja An analogous formula for JQdy may be obtained in a similar manner. Adding the two, we find the formula (29) I Pdx -\-Qdy= f lP<t>'(t) + Qxif'mdt, Jab Ja which 18 the formula for change of variable in line integrals. Of course, if the arc AB ia composed of several portions of different curves, the functions 4>(t) and \l/{t) will not have the same form along the whole of A B, and the formula should be applied in that to each portion separately. IV. $H4] IMPROPFU AVT) ITVK TVTKnnAT.^i 187 94. Area of a closed curv< :y deiiueU the area of a portion of the ])lunu Ix>uii(h / /i, & straight line which does not out that arc, and the two perpendiculars AA^t BBq let fall from the points A and B upon the straight line (§§ G5, 78, Fig. 9). Let us now consider a continuous closed curve of any shape, fay which we shall understand the locus described by a point M whose coordinates are continuous functions x =/(/), y = 4*{0 ^^ * param- eter t which assume the same values for two values to and 7* of the parameter t. The functions /(t) and ^(/) may have several distinct forms between the limits t^ and 7'; such will be the case, for instance, if the closed contour C be composed of portions of several distinct curves. Let A/© , 3/i , 3/, , • • • , 3/< _ , , i/< , • • • , A/, „ , , iV© (Iriiote points upon the curve C corresponding, respectively, to the vahu's /o» Ui fij '•'■> ^i-\i ^ii ••» ^w-u ^ of the parameter, which increase from ^o to 7\ Connecting these points in order by straight lines, we obtain a polygon inscribed in the curve. The limit approatrhed by the aiea of this polygon, as the number of sides is indefinitely increased in such a way that each of them approaches zero, is called the area of the closed curve (\* This definition is seen to agree with that given in the particular case treated above. For if the polygon AqAQ^Q^-- BBqAq (Fig. 9) be broken up into small trapezoids by lines parallel to .-1.40, the area of one of these trapezoids is (ar^ - x,_ ,) [/(x,) -f /(x,_ ,)]/2, or (x^ - x,_,)/(^<), \shere ^, lies between x<_i and a:^. Hence the area of the whole polygon, in this special case, approaches the definite integral //(x)rfx. Let us now consider a closed curve C which is cut in at most two points by' any line parallel to a certain fixed direction. Let us choose as the axis of y a line parallel to this direction, and as the axis of a; a line perpendicular to it, in such a way that the entire curve C lies in the quadrant xOy (Fig. 16). The points of the contour C project into a segment ah of the axis Ox, and any line parallel to the axis of y meets the contour C in at most two points, t/i, and m,. Let yx = ^j(x) and y, = ^s(x) be the equations of the two arcs AniiB and Am^By respectively, and let us suppose for simplicity that the points A and B of the curve C which project into a and b are taken as two of the vertices of the * It Is Bappo0«d, of course, that the curve under oonsideraUon has no double point, and that the Hidtv of the polygon have been chosen so snudl that the polygon itself has no double point. 188 DEFINITE INTEGRALS [IV. §94 polygon. The area of the inscribed polygon is equal to the differ- ence between the areas of the two polygons formed by the lines Aa, abf bB with the broken lines inscribed in the two arcs Am^B and AntxBy respectively. Passing to the limit, it is clear that the area of the curve C is equal to the difference between the two areas bounded by the contours AmtBbaA and Am^BbaA, respectively, that is, to the difference between the corresponding definite in- tegrals ' il/2(x)dx — / il/i(x)dx. a %J a These two integrals represent the curvilinear integral Jyt^a; taken first along Am^B and then along AmiB. If we agree to say that the contour C is described in the positive sense when an observer standing upon the plane and walking around the curve in that sense has the enclosed area constantly on his left hand (the axes being taken as usual, as in the figure), then the above result may be expressed as follows : the area 12 enclosed by the contour C is given by the formula Fia. 16 (30) O = — I ydxy J(C) where the line integral is to be taken along the closed contour C in the positive sense. Since this integral is unaltered when the origin is moved in any way, the axes remaining parallel to their original positions, this same formula holds whatever be the position of the contour C with respect to the coordinate axes. Let us now consider a contour C of any form whatever. We shall suppose that it is possible to draw a finite number of lines connecting pain of points on C in such a way that the retoltiDg Bubcontours are each met in at most two points by any line parallel to the y axis. Such is the case for the region bounded by the contour C in Fig. 1 6, which we may divide into three subregions bounded by the contours amba, abndcqa, cdpc, by means of the Fio. 16 iv,§ufl] IMPKOPKK AND LINE INTEGRALS 189 traDBversaU ab and cd. Applying the preceding formula to each of these subregions and adding the results thus obtained, the line integrals which arise from the auxiliary lines ab and ed cancel each otlier, and tlie area bounded by the closed curve C is still given by the line integral — jy djt taken along the contour C in the positive sense. Similarly, it may be shown that this same area is given by the formula (31) Q=fxdyi and finally, combining these two formulae, we have (32) " = 9 / xdy-ydxy where the integrals are always taken in the positive sense. This last formula is evidently independent of the choice of axes. If, for instance, an ellipse be given in the form ar = acos^, y = 6sin^, its area is 1 r" O = - I a6(cos*^ 4- sin*<) dt = irab. 95. Area of a curve in polar coordinates. Let us try to find the area enclosed by the contour 0AM HO (Fig. 17), which is composed of the two straight lines 0.1, OBy and the arc AMB^ which is met in at most one point by any radius vector. Let us take O as the pole and a straight line Ox as / the initial line, and i let p = /(w) be the V equation of the arc A MB. Inscribing a polygon in the arc AMB^ with A and B as two of the vertices, the area to be evaluated is the limit of the sum of such triangles as OMM\ But the area of the triangle OMM^ ia Flo. 17 ^ p(/) H- Ap) sin A« = Awf^ 4-<j, 190 DEFINITE INTEGRALS [IV, §95 where < approaches zero with Aw. It is easy to show that all the quantities analogous to c are less than any preassigned number rj provided that the angles A(o are taken sufficiently small, and that we may therefore neglect the term cAw in evaluating the limit. Henoe the area sought is the limit of the sum 2/)* Aw/2, that is, it is equal to the definite integral 2X where <i>i and ta^ are the angles which the straight lines OA and OB make with the line Ox. An area bounded by a contour of any form is the algebraic sum of a certain number of areas bounded by curves like the above. If we wish to find the area of a closed contour surrounding the point O, which is cut in at most two points by any line through 0, for example, we need only let <u vary from to 27r. The area of a con- vex closed contour not surrounding (Fig. 17) is equal to the dif- ference of the two sectors 0AM BO and OANBO, each of which may be calculated by the preceding method. In any case the area is represented by the line integral U' taken over the curve C in the positive sense. This formula does not differ essentially from the previous one. For if we pass from rectangular to polar coordinates we have X = p cos o>, y = p sin CD, dx = COS to dp — p sin to du), dy = sin <adp -^ p cos o> doty X di/ — 7/ dx = p^do). Finally, let us consider an arc A MB whose equation in oblique coordinates is y =f(x). In order to find the area bounded by this arc AM By the x axis, and the two lines AA^, BBq, which are parallel to the y axis, let us imagine a polygon inscribed in the arc A MB, and Icfc ua break up the area of this polygon into small trapezoids by lioet parallel to the y axis. The area of one of these trapezoids is ^"-'K'-^'"\ ^,-^...)sine, IV, §96] IMPROPER AND LINE INTEGRALS 191 which may be written in the form (x^_^— Xt)/(fi)A\n$, where ^ lies in the interval (x^.d x^). Hence the area in question is equal to the definite integral 8in^ / /{x)dXf where Xq and A' denote the abscisssB of the points A and Bj respectively. It may be shown as in the similar case above that the area bounded by any closed contour C whatever is given by the formula -y I xdy-ydx. Note. Given a closed curve C (Pig. 16), let us draw at any point M the portion of the normal which extends toward the exterior, and let a, /3 be the angles which tliis direction makes with the axes of X and y, respectively, counted from to tt. Along the arc A mj B the angle /3 is obtuse and cte = — dscosp. Hence we may write Ji/dx = — j y cos fids. {Am^B) J Along BytiiA the angle fi is acute, but dx is negative along Bm^A in the line integral. If we agree to consider ds always as positive, we shall still have dx = — ds cos fi. Hence the area of the closed curve may be represented by the integral / ycospdSf where the angle fi is defined as above, and where d* is essentially positive. This formula is applicable, as in the previous case, to a contour of any form whatever, and it is also obvious that the area is given by the formula / r cos It ds. These siatements are absolutely independent of the choice of axea. •6. ValM of the intesral /xdy - ydx. Ii Is nmlural to inquire what will be represented by the integral fzdy — ydz, taken nv«»r ^nv ..irvn whatever, doeed or oncloeed. 192 DEFINITE INTEGRALS [IV, §97 Let us consider, for example, the two closed curves OAOBO and ApBqCrAtBtCuA (Fig. 19) which have one and three double points, respec- tively. It is clear that we may replace either of these curves by a combination of two closed curves without double points. Thus the closed contour OAOBO is equivalent to a combination of the two contours OAO and OBO. The integral taken over the whole contour is equal to the area of the portion OAO less the area of the portion OBO. Likewise, the other contour may be replaced by the two closed curves ApBqCrA and AsBtCuA, and the integral taken over the whole con- tour is equal to the sum of the areas of ApBsA, BtCqB, and ArCuA , plus twice the area of the portion AsBqCuA. This reasoning is, moreover, general. Any closed contour with any number of double points determines a certain number of partial areas <ri, <r2, • • •, <rp, of each of which it forms all the boundaries. The integral taken over the whole contour is equal to a sum of the form miffi + m^ff^ 4- • • • + mpo-p, where mi, m^, • • •, wip are positive or negative integers which may be found by the following rule : Given two adjacent areas <r, <r', separated by an arc ab of the contour C, imagine an observer walking on the plane along the contour in the sense determined by the arrows ; then the co^cient of the area at his left is one greater than that of the area at his right. Giving the area outside the contour the coeflB- cient zero, the coefficients of all the other portions may be determined successively. If the given arc AB'\& not closed, we may transform it into a closed curve by joining its extremities to the origin, and the preceding formula is applicable to this new region, for the integral fxdy — ydx taken over the radii vectores OA and OB evidently vanishes. V. FUNCTIONS DEFLN^ED BY DEFINITE INTEGRALS 97. Differentiation under the integral sign. We frequently have to deal with integrals in which the function to be integrated depends not only upon the variable of integration but also upon one or more other variables which we consider as parameters. Let f(x, a) be a continuous function of the two variables x and a when x varies from r^U) X and a varies between certain limits ctq and ai. We proceed to study the function of the variable a which is defined by the definite integral where a is supposed to have a definite value between «„ and aj, and where the limits x^ and X are independent of a. Iv.iVT] FUNCTIOIJS DEFINED BY INTEGRALS 198 We have then (33) F(a + Aa) - F(a) = f [/(x, a + Aar) -/(x, a)] dr. Since the function /[x, a) is continuous, this integrand may be made less than any preassigned number c by taking Aa sufficiently small. Hence the increment ^F{a) will be less than €\X — x^\ in absolute value, which shows that the function F(a) is continuous. If the function f(Xf a) has a derivative with respect to a, let us write f(x, a -f Aa) ^/(x, a) = Aa [/. (x, a) + t] , where c approaches zero with A<r. Dividing both sides of (33) by Aa, we find F(a 4- Aa) - F(a) f\ . ^ C' . and if 17 be the upper limit of the absolute values of c, the absolute value of the last integral will l)e less than ri\X — Xq\. Passing to the limit, we obtain the formula In order to render the above reasoning perfectly rigorous we most show that it is possible to choose Aa so small that the quantit}' c will be less than any preassigned number rj for all values of r l^etween the given Jimits Xq and A'. This condition will certainly be satisfied if the derivative /«(x, a) itself is continuous. For we have from the law^of the mean /(x, a 4- Aa) -/(x, a) = Aa/, (x, a -f tf Aa), < tf < 1, and hence €=M^,^'he^a)-/,(x,a), If the function/, is continuous, this difference c will be less than if for any values of x and a, provided that i Aa| is less than a properly chosen positive number h (see Chapter VI, $ 120). I^t us now suppose that the limits A' and x, are themselves func- tions of a. If A A' and Aj-q denote the increments which correspond to an increment Aa, we shall have 194 DEFINITE INTEGRALS tIV,§97 Jan, /(a;,a + Aar)<i:aj I /(x, a4-AQ:)rfa;; or, applying the first law of the mean for integrals to each of the last two integrals and dividing by Aa, F(a-\-Aa)-F(a) ^ C "" f(x, a + Aa) - f(x, a) ^^ Aa J^^ Aa AX -\- — f(X -^ e AX, a + Aa) Aor -■^f(x,-^e'Ax,,a-\-Aa). As Aa approaches zero the first of these integrals approaches the limit found above, and passing to the limit we find the formula which is the general formula for differentiation under the integral sign. Since a line integral may always be reduced to a sum of ordinary definite integrals, it is evident that the preceding formula may be extended to line integrals. Let us consider, for instance, the line integral F(a) = I P{xy y,a)dx-ir Q(x, ij, a)dy Jab taken over a curve A B which is independent of a. It is evident that we shall have Jai Pa(Xy y, a)dx + Q„{x, y, a)dy, where the integral is to be extended over the same curve. On the other hand, the reasoning presupposes that the limits are finite and that the function to be integrated does not become infinite between the limits of integration. We shall take up later (Chapter VIII, f 175) the cases in which these conditions are not satisfied. IV, §98] FFVCTIONS PFFTVKD BY INTF.ORAI^i 196 The foiiiiuhi ^^35) is frequuutly i integrals by reducing them to oth« i i u lated. Thus, if a is poeitiye, we have /■ €1x1 X -J- — = -p arc tan -7= > whence, applying the formula (34) n — 1 times, we find (-l).-.1.2...(»-l)jr'^. = £^(^are,a„^). 9S. Examplet of discontinuity. If the conditions imposed are not satisfied for all valaes between the liinitH of integration, it may happen that the definite Inte- gral defines a discontinuous function of the parameter. Let us consider, for example, the definite integral »/ V r sinadx F{a) ■i>- 2x cos a + 2^ This integral always has a finite value, for the roots of the denominator are imaginary except when <r = krt, in which case it is evident that f\a) — 0. Sup- posing that sin a ;e and making the substitution z = cos a -^ t sin a, the indefi- nite integral becomes - — -= I ; — - =arctan<. 1 - 2xcosar + x« J l + (^ Hence the dt finite integral F{a) has the value ^ /I — cosaX ^ /-I — cosa\ arc tan ( — ; ) — arctani ) « \ sina / \ sina / where the angles are to be taken between - x/2 and x/2. But 1 — cosa — 1 — cosa sin or sin a = -1, and hence the difference of these angles is ± n/2. In order to determine the sign uniquely we need only notice that the sign of the integral is the same as that of sin a. Hence F(a) = db n/2 according as sin a is positive or negative. It follows that the function F{a) is discontinuous for all values of (i: of the form At. This result does not contradict the above reasoning in the least, however. For when x varies from — 1 to -f 1 and a varies from — c to + t, for example, the function under the integral sign assumes an indeterminate form for the sets of values nr = 0, x = - 1 and a = 0, x = + 1 which belong to the region in qnee- tion for any value of 9. It would be easy to give numerous examples of this nature. A|;ain. consider the integral * * sin mx I dx. 196 DEFINITE INTEGRALS [IV, §99 the snbetltution mx = y, we find X"*" * sin mx , , r — -dy, where the aign to be taken is the sign of m, since the limits of the transformed integral are the same as those of the given integral if m is positive, but should be interehanged if m is negative. We have seen that the integral in the second member is a positive number JV (§ 91 ). Hence the given integral is equal to ± ^ according as m is positive or negative. If m = 0, the value of the integral is rero. It is evident that the integral is discontinuous for m = 0. VI. APPROXIMATE EVALUATION OF DEFINITE INTEGRALS 99. Introduction. When no primitive of f(x) is known we may- resort to certain methods for finding an approximate value of the definite integral Cf(x) dx. The theorem of the mean for integrals furnishes two limits between which the value of the integral must lie, and by a similar process we may obtain an infinite number of others. Let us suppose that <f>{x) <f(x) < i{/(x) for all values of x between a and b (a< b). Then we shall also have <l>(x)dx< I f(x)dx< I \l/(x)dx. U a %J a If the functions ^{x) and ^{x) are the derivatives of two known functions, this formula gives two limits between which the value of the integral must lie. Let us consider, for example, the integral =x- dx Now Vl-x* = Vl-a;« Vl + x^, and the factor VlT^ lies between 1 and V2 for all values of x between zero and unity. Hence the given integral lies between the two integrals dx Jo Vl-a;«' V2 Jo • that Ib, between 7r/2 and 7r/(2V2). Two even closer limits may- be found by noticing that (l-\-x^)-^^^ is greater than 1 — a72, which results from the expansion of (1 -f u)-^^^ by means of Taylor's series with a remainder carried to two terms. Hence the integral / is greater than the expression r dx 1 r Jo vr=^ 2jo x^dx IV. §wj APPROXIMATE EVALUATION' 197 Thu stMoiid of these integrals has the value ir/4 (§ 105) ; hence / lies between 7r/2 and 3 tt/S. It is evident that the preceding methods merely lead to a rough idea of the exact value of the integral. In order to obtain closer approximations we may break up the interval (a, b) into smaller subintervals, to each of which the theorem of the mean for inte- grals may be applied. For deiiniteness let us suppose that the function f(x) constantly increases as x increases from a to b. Let us divide the interval (a, b) into n equal parts (b — a = nh). Then, by the very definition of an integral, j^f{x)dx lies between the two sums ' = Aj/(a) +/(«4-/0 4- ••+/[«+ (n-l)A]{, S = h\f(a + h) -\-f(a + 2A) + • +/(« -f nh)\, I f we take (S -h *)/2 as an approximate value of the integral, the error cannot exceed |5-«|/2 = |[(6 - a)/2 w][/(*)-/l(a)]|. The value of (.s -f «)/2 may be written in the form jt ( /['')+ /(« + *) , /(« + /0+/^» + 2A) ^ (2 ^ "^ 2 i' Observing that j/(a -f »A) +/[a + (» 4-l)A]jA/2 is the area of the trapezoid whose height is h and whose bases SLref(a + ih) and /{a + ih -H A), we may say that the whole method amounts to replacing the area under tlie curve y = f(x) between two neighbor- ing ordinates by the area of the trapezoid whose bases are the two ordinaties. This method is quite practical when a high degree of approximation is not necessary. Let us consider, for example, the integral i: dx Taking 7) = 4, we find as the approximate value of the integral and the error is less than 1/16 = .0625.* This gives an approxi- mate value of IT which is correct to one decimal place, — 8.1311 • • •. • Found from Uie formula \a — »\/% In fact, Um error is abont .00880, the exaot value being it/\.—1iUi»: 198 DEFINITE INTEGRALS [IV, §100 If the function f{x) does not increase (or decrease) constantly as X increases from a to i, we may break up the interval into sub- intervals for each of which that condition is satisfied. 100. Interpolation. Another method of obtaining an approximate value of the integral f^f(x) dx is the following. Let us determine a parabolic curve of order n, y = 4>(x) = tto + ai* H h a„a5", which passes through {n -f- 1) points Bq^ B^ •••, B„ of the curve y —f{x) between the two points whose abscissae are a and b. These points having been chosen in any manner, an approximate value of the given integral is furnished by the integral f^<f>(x)dx, which is easily calculated. Let (xof yo), («n yi)i • • > (^«> ^n) be the coordinates of the (n+l) points Boy Biy •••, B„. The polynomial <l>(x) is determined by Lagrange's interpolation formula in the form ^(x) = yoXo+yi^i+-- + y,X, + ... + y„Z„, where the coefficient of y,- is a polynomial of degree n, * (x, - Xo) • • • i^i - Xi_{) (X, -X,^,)-'- (Xi - x„) ' which vanishes for the given values cco, aji, • •, a;„, except for a: = a;,., and which is equal to unity when a; = «,. Hence we have ' <^(x)rfx = y Vil X,dx. a f^Q Ja The numbers x^ are of the form «o = a + ^o(* - a), aji = a H- B^{b - a), . . ., «„ = a + BJh - a), where < do < ^i < • • • < d„ 5 1. Setting x = a ■\- {h - a)t, the ap- proximate value of the given integral takes the form (36) (^-a)(^oyo + /^iyi+-+A',y.), where K^ is given by the formula /r,= r (^-^o)''(^-<?.-i)(^-g...,)..(^-gj Jo (^i-^o)--(^.-^,-OW-^.>i)-(^.-^„) If we divide the main interval (a, h) into subintervals whose ratios are the same constants for any given function /(a;) whatever, the numbers do, ^i, • ■ , d,, and hence also the numbers A'.., are inde- pt'udent of /(x). Having calculated these coefficients once for all, I\,ii.Mj APPROXIMATE KVALUATION 199 it only remains to replace j/ot !/if "t Vm^ ^^^i' retpeotiTe yalues in the formula (36). If the curve f(x) whoee area is to be evaluated is given graph- ically, it is convenient to divide the interval (a, b) into equal part*, and it is only necessary to measure certain equidistant ordinates of this curve. Thus, dividing it into halves, we should take $^ = 0, di = 1/2, d, = 1, which gives the following formula for the approxi- mate value of the integral : b-a /=-g— Cyo + 4y, 4-yi). Likewise, for n = 3 we find the formula / = ^-^ (yo + 3y» + 3y, -f y.), and for n = 4 / = -^'^ (7yo + 32y, + 12y, + 32y. + Ty,). The preceding method is due to Cotes. The following method, due to Simpson, is slightly different. Let the interval (a, b) be divided into 2n equal parts, and let yo» yn y«» ••» y*. be the ordi- nates of the corresponding points of division. Applying Cotes* formula to the area which lies between two ordinates whose indices are consecutive even numbers, such as y© and y,, y, and t/t, etc., we find an approximate value of the given area in the form / = -^"^[(yo -f 4yi + y,) + (y, 4- 4y, + yO -^ • • • + (y«.-f + 4y»,_, + y^)], whence, upon simplification, we find Simpson's formula: b- a ^ = -^ [yo 4- yx., T - v.y3 -r y« -r • • • 4- t/u^t) 4-4.',/. -^,/. 4-...4.y^.,)]. 101. Gauss' method, in dauss m* iii<>d other values are assigned the quantitie.s $^. The arguiiuiit is :is follows: Suppose that we ran find polynomials of increasing <: iiich differ less and less from the given integrand /(x) in luu interval (a, b). Suppose, for instance, that we can write /(x) = ao + <»i* + a,a;« + • • • -h «tie.i«*'"* + Ru(')^ where the remainder R^i^) is ^^^^ ^^^^^^ ^ fixti! number < for all 200 DEFINITE INTEGRALS [IV. §101 yalues of x between a and h* The coefficients a,- will be in gen- eral unknown, but they do not occur in the calculation, as we shall see. Let x©, x^, ••, ar^-i be values of x between a and 6, and let <^(x) be a polynomial of degree n — \ which assumes the same values as does f{x) for these values of x. Then Lagrange's inter- polation formula shows that this polynomial may be written in the form jii-i *(*) = X ''-^'-(^) ■*■ ^a.(a^0)^0 W + • • • + ^2,.(^»-l) *n-l W, where ^^ and 4^^ are at most polynomials of degree n — 1. It is clear that the polynomial <^,„(x) depends only upon the choice of 3Co> ^i»"*> ^11 -1- ^^ ^'^ other hand, this polynomial <^«(ic) must assume the same values as does ic"* ioY x — x^^ x — x-^, • • •, x — x^_^. For, supposing that all the a's except a^ and also Tt^^^ix) vanish, f{x) reduces to o-^x"* and f\t{x) reduces to a^i^^{x). Hence the difference x"* — «/>«(«) must be divisible by the product P^{x) = {x- Xo) (x- Xi) ■■■ (x- iC„_i). It follows that X"* — <f>„,(x)= P„Q,n_,,(x), where Q,„_„(aj) is a poly- nomial of degree m — n, if m > 7^ ; and that x"" — (f>j^(x) = it m <n — 1. The error made in replacing J^ f(x) dx by J^ <^ {x) dx is evidently given by the formula (37) Va, I [X- - ^^{x)\ dx + / R,,(x)dx n-1 ^6 t = *Ja The terms which depend upon the coefficients a^, a-^, • • •, «-„ _i vanish identically, and hence the error depends only upon the coefficients ^mt ^u-^M "} ^ti-\ ^^d *he remainder R2„(x). But this remain- der is very small, in general, with respect to the coefficients ^nt ^'n + n •••> ^2«-i- Hence the chances are good for obtaining a high degree of approximation if we can dispose of the quantities *o» ari, •••, x,_, in such a way that the terms which depend upon "•» *■ + !» •••» ocu-i also vanish identically. For this purpose it is necessary and sufficient that the n integrals X*> /»«• •»& • ThU Is A property of any function which is continuous in the interval (a, 6), to a theorem due to Wolorutrass (see Chapter IX, § 199). IV, 1102] APPROXIMATE EVALUATION 201 should yanish, where Q< is a polynomial of degree i. We have already seen (§ 88) that this condition is satisfied if we take P. of the form It is therefore sufficient to take for x^, x^, -", x..| the n roots of the equation /'. = 0, and these roots all lie between a and b. We may assume that a = — 1 and b = + 1, since all other caies may be reduced to this by tlie substitution x = (b -\- a) / 2 -\-t(h — a) /2. In the special case the values of ar^, Xj, •••, x,_, are the roots of Legendre's polynomial A\. The values of these roots and the values of A', for the formula (36), up to n = 5, are to be found to seven and eight places of decimals in Bertraud's Traite de Caleul integral (p. 342). Thus the error in Gauss' method is f Ru{')dx-'^R^{x,) f\(x)dx, where the functions ^.(ar) are independent of the given integrand. In order to obtain a limit of error it is sufficient to find a limit of R^(x)t that is, to know the degree of approximation with which the function f(x) can be represented as a polynomial of degree 2n — 1 in the interval (a, h). But it is not necessary to know this polynomial itself. Another process for obtaining an approximate numerical value of a given definite integral is to develop the function /(or) in series and integrate the series term by term. We shall see later (Chapter VIII) under what conditions this process is justifiable and the degree of approximation which it gives. 108. Anuler'a pUnimeter. A great many machines have been invented to measure mechanically the area bounded by a cloeed plane curve.* One of the most ingenious of these is Amsler^a plauimeter, whose theory affords an interest- ing application of line integrals. Let us consider the areas Ai and Aj bounded by the carves described by two points A I and ^s of a rigid straight line which moves In a plane in any manner and finally returns to its original position. Let (Xi, yi) and (x^, y,) be the co^r dinates of the points ^i and .^ti respectively, with respect to a set of rectaugn- Ur axes. Let / be the distance AiAtn and the angle which Ai At makes with * A dMcription of these instrumenu is to be foood in a wwk by Abdank- Abakanowir/ : Ia^ inUffrapKes, la courts inUgmU H ssf oppHoiUion» (Gaathie^ VilUre, 188G). 202 DEFINITE INTEGRALS [IV, §102 the positive x axis. In order to define the motion of the line analytically, Xi, j/i, and 6 must be supposed to be periodic functions of a certain variable parameter t which resume the same values when t is increased by T. We have Xa = Xi + i cos^, ys = Vi + ' sin e, and hence Xtdyt - y«dxi = Xidyi - j/idxi + Pdd + ^(cos^dyi — 8in<?dxi -\-XiCosddd -\-yiBmede). The areas Ai and Aj of the curves described by the points Ai and ^j, under the general conventions made above (§ 96), have the following values : Ai = - fxidyi - yidxu ^2 = J^^V^ ~ V^^- Hence, integrating each side of the equation just found, we obtain the equation At = Ai + - Cde + - jcosddyi — am0dxi+ j{xi coBd + yisin tf) dtf , where the limits of each of the integrals correspond to the values to and to+ T of the variable t. It is evident that fdd = 2K7e, where K is an integer which depends upon the way in which the straight line moves. On the other hand, integration by parts leads to the formulae Jxi cos ^d^ = xi sin d — fsin ddxi^ Jyi8medd = — yicoad+ jcosddyi. But xi sin and yi cos 6 have the same values tor t = to and t = to-\- T. Hence the preceding equation may be written in the form As = Ai + Ejtl^ + I icoaddyi — sintf dxi. Now let a be the length of the arc described by Ai counted positive in a certain sense from any fixed point as origin, and let a be the angle which the positive direction of the tangent makes with the positive x axis. Then we shall have cos^dyi -sin^dxi = (sinacostf - sintfcosa)d« = sinFds, where V is the angle which the positive direction of the tangent makes with the potitive direction AiAt of the straight line taken as in Trigonometry. The pnoeding equation, therefore, takes the form (88) Aa = Ai + KnP + iJsmVda. Similarly, the area of the curve described by any third point At of the straight line is given by the formula («<>) A| = Ai + Kicn + rJsinFds, li. Eliminating the unknc , we find the formula rA, - < A, = (r - t) Ai + K7cU\l - i'), where r is the disUnce AiAt. Eliminating the unknown quantity /sinFda totwaea tbase two equations, we find the formula IV,Jlt«] APPROXIMATE EVALUATION 208 which may be written in the form (40) Ai(23) + A,(:)l) + A,(12) + Jrjr(l2)(28)(81) = 0, where (ik) denotes the diHtance between the polnu Ai and il* (<, Jk s 1, S, S) ukeii witli itii proper sign. A« an appUoation of this formuia, let na oooalder a etraight line .^i^s of length (a + 6), whose extreroitiee Ax and At deaeribe the •ante doaed convex curve C. The point ^Ig, which divkiea the line into aeg- menti of length a and 6, describes a closed curve C which lies wholly indde C. In this case we have A,= A|. (12) = o + 6, (28) = -6, (81) = -a, ir = l; whence, dividing by a + 6, Ai — At = xab. But Ai - At iB the area between the two curves C and C. Hence this area ia independent of the form of the curve C. This theorem is due to Holditch. If, instead of eliminating jBinVda between the equations (88) and (89), we eliminate Ai, we find the formula (41) At = A, + Kx(V* _ p) + (r _ f)JsinFd». Amsler's planimeter affords an application of this formula. Let AiA^At be a rigid rod joined at At with another ro<l OAt- The point O being fixed, the point ^t« to which is attached a sharp pointer, is made to describe the curve whose i is sought. The point At then describes an arc of a circle or '2^ A'^* an entire circumference, accord- ing to the nature of the motion. In any case the quantities As, £*, Z, V are all known, and the area At can be calculated if the in- tegral /sin Vds, which is to be taken over the curve C\ described by the point Ax, can be evaluated. This end .^i carries a graduated ^"^ circular cylinder whose axis coin- cides with the axis of tlie rod i4ti4s, and which can turn about this I^et us consider a small displacement of the rod which carries Ax At At Into the position A'\ A%Ai. I>et (^ be the intersection of these straight linea. About Q as center draw the circular arc A\a and drop the perpendicular ^(P from A\ upon AxAf We may imagine the motion of the rod to consist of a sliding along its own direction until Ax comes to a, followed by a rotation about ^ which brings alo A{. In the first part of this process the cylinder would slide, with- out turning, along one of ita generators. In the second part the roution of the cylinder is measured by the arc aAi. The two ratios aA'x/A'xP and ill P/arc A\A{ approach 1 and sin V, respectively, as the arc ^(^i approaches aero. Hence a^( = As (sin V + «), where « approaches zero with As. It follows that the total rotation of tlie cylinder is proportional to the limit of the sum £A«(Min r 4- (), that is, to the integral /sinKcCa Henoe the meaauremeDt of this roUtion is sufficient for ttie determination of the given 204 DEFINITE INTEGRALS [iv, Exs, EXERCISES 1. Show that the sum 1/n -f l/(n + 1) + • • + l/2n approaches log 2 as n inoresMS indefinitely. [Show that this sum approaches the definite integral J^ [1/(1 + x)] dx as its limit.] 2. As in the preceding exercise, find the limits of each of the sums n« + l n« + 2!' n2 + (n-l)» 1 +^.i^ + ...+ 1 Vn«-1 Vn2-22 V»2 - (n - 1)2 bj connecting them with certain definite integrals. In general, the limit of the sum ^'t>{hn), as n becomes infinite, is equal to a certain definite integral whenever 0(i, n) is a homogeneous function of degree — 1 in i and n. 3. Show that the value of the definite integral //^^ log sin x dx is -(jr/2)log2. [This may be proved by starting with the known trigonometric formula sm-sm sm^ '— = , n n n 2'»-i or else by use of the following almost self-evident equalities : WW — r*logsinxdx = I ^ logcosxdx = - I log/ idx.l Jo Jo 2j, V 2 / ■■ 4. By the aid of the preceding example evaluate the definite integral IT Jlx jtanxdx. i, V 2/ 6. Show that the value of the definite integral dx r'logOjfx) Jo 1 + ^" li(jr/8)log2. [Set X E un and break up the transformed integral into three parts.] 6*. Sfaluate the definite integral J log(l- 2acosx + a^(2x. [POISSON.] IV. Kxi.] EXERCISES 206 [ Dividing the interval from to ir into n equal parts and applying a well^nown formula of trigonometry, wa are led to aeek the limit of the eTpwton ?Iog[^(a--I)] n La + 1 -• aa n beoomee infinite, if a lies between — 1 and -f 1, this limit la »!ro. If a« > 1, it is jr log a*. Compare f 140.] 7. Show that the value of the definite integral i: slnxdg VI- 2arcos2 + afl where oc is positive, is 2 if or < 1, and is 2/a if a > 1. 8«. Show that a necessary and sufficient condition that /(z) ihoald be int»- f^rable in an interval (a, b) is tliat, corresponding to any preassigned number e« a subdivision of the interval can be found such that the difference 5 — f of the corresponding sums 8 and a is less than e. 9. Let/(z) and 0(x) be two functions which are continuousan the interval (cu 6)* and let (a, Zi, x«, • • , 6) be a method of Subdivision of that interval. If (,, ir^ are any two values of x in the interval (x,_f, x,), the sum ^/{^i) <P{vt'^ (Xi — Xf_i) approaches the definite integral f^/{x) 0(z) dz as its limit. 10. Let/(x) be a function which is continuous and positive in the interval (a, b). Show that the product of the two definite integrals !>''''• £m \a a minimum when the function is a constant. 11. Let the symbol /'> denote the index of a function (§ 77) between So and X\. Show that the following formula holds: where e = + 1 if jyxa] «» anu _nj\} <0, « = — 1 if /(zo) <0 and /(zi)>0, and « = if /(xo) and /(zO have the same eign. [Apply the laAt formula in the second paragraph of f 77 to each of the ftino- Uons/(x) and l//(x).] 12*. Let U and V be two polynomials of degree n and n — 1, reepeetlvely, which are prime to each other. Show that the index of the rational fraction V/U between the limits - ao and + oo is equal to the difference between the number of imaginary roots of the equation V + t'T = in which the ooeflkitont of i is positive and the number in which the coefficient of t is negative. [Hbrmitk, BulUiin de la 8oei^ fMiM^maUque, Vol. VII, p. 128.] 13*. Derive the second theorem of the mean for iniegrala by integrmtioii 1^ I»arts. 206 DEFINITE INTEGRALS [IV,Exs. rLet/(x) and ^(z) be two functions each of which is continuous in the inter- Tal (a, b) and the ftrat of which, /(z), constantly increases (or decreases) and bM a oonllnaoui derivaUve. Introducing the auxiliary function *(z)=fy{x)dx and integrating by parU, we find the equation J /(x) 0(«) dx = f(b) *(6) - fHx) *(x) dx . 8lDoe/'(x) always has the same sign, it only remains to apply the first theorem of the mean for integrals to the new integral.] 14. Show directly that the definite integral fxdy-ydx extended over a eloMd eontoor goes over into an integral of the same form when the axes are ropUoed by any other set of rectangular axes which have the same aspect. 15. Given the formula Jr.6 1 cosXxdx = - (sin X6 - sin Xa), a X traloate the integrals r x»p + ^sinXzdx, fx^PcosXxdx. 16. Let us associate the points (x, y) and (x\ y') upon any two given curves C and C, respectively, at which the tangents are parallel. The point whose codrdinates are Xi = px + qx\ yi=py -\- qy% where p and q are given constants, describes a new curve Ci. Show that the following relation holds between the co t TM p onding arcs of the three curves : 8i=±p8± qsf. 17. Show that corresponding arcs of the two curves x = tf\t)-f{t) +0'(«), ^, (x' = r(0-/(«) -<P'{th V =nt) - Wit) + 0(0, w =nt) + w{t) - 0(«) hare the same length whatever be the functions /(f) and <f>{t). 18. From a point If of a plane let us draw the normals MPi, • • • , MPn to a given carves Ci, Cs, • • •, Cn which lie in the same plane, and let U be the distance MPi. The locus of the points M, for which a relation of the form F(l|, ttt • • •♦ M = holds between the n distances i,, is a curve r. If lengths pfoportioaal to tF/dU be laid off upon the lines MPi, respectively, according to * <liftitte eoBTentlon as to ilgn, show that the resultant of these n vectors gives tiM dirtetion of the normal to r at the point M. Generalize the theorem for in 19. l«t C be any cloeed curve, and let us select two points p and p' upon the U> C at a point m, on either side of m, making mp = mp\ Supposing that the distance mp varies according to any arbitrary law as m describes the C, dbow that the points p and p' describe curves of equal area. Discuss where mp la constant. i\ , Exs] EXERCI8E8 207 20. Given any closed conTex cunre, let ai draw a panOal eai-ft bj UjiBf oA a constant length I upon the norm&la to the given cture. Show tliAt Um m«a bet>^en the two curves U equal to ± ir(> + at, wh«r« • ia the laogtli o( tba 21. Let C be any closed curve. Show that the loena of the poinU A, for which the corresponding pedal has a constant area, is a circle wboas osai« ii fixed. [Take the equation of the curre C in the tangntial form zcost-f y sine =/(<).] 22. Let C be any closed curve, Ci its pedal with respect to a point A, and C| the locus of the foot of a perpendicular let fall from A upon a normal to C. Show that the areas of these three curves satisfy the relation A = Ai - A^. [By a property of the pedal (§ 36), if /> and m are the polar coordinates of a point on Ci, the coordinates of the corresponding point of Cs are / and « + r/2, and those of the corresponding point of C are r = Vf^ + p'* and ^ = m 4- are tan Z*/^.] 23. If a curve C rolls without slipping on a straight line, every point A whkh is rigidly connected to the curve C describes a curve which is called a rMictts. Show that the area between an arc of the roulette and its base Is twice the area of the corresponding portion of the pedal of the point A with respect to C. Also show that the length of an arc of the roulette is equal to the loDgth of the corre- sponding arc of the pedal. ro— . i ^BTKIJIBB.J [In order to prove these theorems analytically, let X and T be the eodidl- nates of the point A with respect to a moving system of axes formed of the tangent and normal at a point M on C. Let « be the length of the are OM counted from a fixed point O on C, and let w be the angle between the tanfSDta at and M. First establish the formulse da-\-dX= Ydw, dF + XA# = 0, and then deduce the theorems from them.] 24*. The error made in Gauss* method of quadrature maj be eipvMnd la the form X 2 r I.2.8. -m -1* + lLl.J.-(«ii-l)J' 1 . 2 . . . 2n 2n where { lies between - 1 and +L [Mahsiob, Convtes rtmiuM, 1886.] CHAPTER V INDEFnaXE INTEGRALS We shall review in this chapter the general classes of elemen- tary fonctioDB whose integrals can be expressed in terms of ele- owntary functions. Under the term elementary functions we shall include the rational and irrational algebraic functions^ the exponen- tial function and the logarithm, the trigonometric functions and their inrerses, and all those functions which can be formed by a finite number of combinations of those already named. When the indefinite integral of a function f(x) cannot be expressed in terms of these functions, it constitutes a new transcendental function. The study of these transcendental functions and their classification is one of the most important problems of the Integral Calculus. I. INTEGRATION OF RATIONAL FUNCTIONS 103. General method. Every rational function f(x) is the sum of an integral function E(x) and a rational fraction P(x)/Q(x), where P{x) is prime to and of less degree than Q{x). If the real and imaginary roots of the equation Q{x) be known, the rational frac- tioo may be decomposed into a sum of simple fractions of one or the oiher of the two types A Mx-{- N The fnetions of the first type correspond to the real roots, those of the eeoond type to pairs of imaginary roots. The integral of the integral function E(x) can be written down at once. The inte- grtls of the fractions of the first type are given by the formulae Adx A (^^^ — (m^lXx-ar-^' ifm>l; = ^log(aj-a), if ?;i = l. for lU take of simplicity we have omitted the arbitrary constant C, which htlOBfi on the rigUtrhand side. It merely remains to examine 908 / Adx X — a V.$103] RATIONAL FUNCTIONS S09 the simple fractions which arise from pairs of imigioary rtKHa In order to simplify the corresponding integrmU, let us maW^ ^ substitution a5 = a + /8f, dx^fidt. The integral in question then beoomes J [(^ - «)' + ^"]- " iS"- V (1 + <«)V. ^ and there remain two kinds of integrals : r tdt r dt J (1 + 0-' J (1 + ^-* Since tdt is half the differential of 1 4- ^', the first of these inte- grals is given, if n > 1, by the formula / tdt 1 ff*-« (l + <0"~ 2(»-l)(l + «')— " 2(«-l)[(z-a)« + ^]' or, if n = 1, by the formula The only integrals which remain are those of the type /; dt If n = 1^ the value of this integral is / dt ^ ^ ♦ •^-<' = arc tan ^ = arc tan l+^» P If n is greater than unity, the calculation of the integral may be reduced to the calculation of an integral of the same form, in whioh the exponent of (1 + t^) is decreased by unity. Denoting the inie gral in question by /,, we may write r - C dt __ r i-h^-<» __ r dt _ r ^dt ^--j (i+ty-j (14-0- '"-J (i + o— J(i + <v* From the last of these integrals, taking tdt 1 (l + O" 2(fi-l)(l + r«)-» 210 INDEFINITE INTEGRALS [V,§103 and integrating by parts, we find the formula C i^dt t , _J__ C_Ji__, JiTT^'' 2(n-l)(l4-^*r-* 2(71-1) J (1+^r-^ Substituting this value in the equation for /^, that equation becomes ,=2^^...+ Repeated applications of this formula finally lead to the integral /i = arc tan t. Retracing our steps, we find the formula , (2n-3)(2n-6) .3.1 ^ , , _.,. ^- = (2n-2)(2n-4)...4.2 "^^^""^ + ^(^)- where R(t) is a rational function of t which is easily calculated. We will merely observe that the denominator is (1 + t^y~^, and that the numerator is of degree less than 2n — 2 (see § 97, p. 192). It follows that the integral of a rational function consists of terms which are themselves rational, and transcendental terms of one of the following forms : log (x — a), log [(x — ay -f- )8*], arc tan X Lei OB consider, for example, the integral J[l/(x^ — 1)] dx. The denominator has two real roots + 1 and — 1, and two imaginary roots + » and — t. We may therefore write 1 A , B Cx-{- D x*-l aj-lx + l' l-\-x^ In order to determine A^ multiply both sides by a; — 1 and then set X = 1. This gives A = 1/4, and similarly B = — 1/4. The iden- tity assumed may therefore be written in the form -1— 1/_J L\ x*-l 4Va;-l x + lj Cx -\-D l-{-x^ or, simplifying the Jeft-hand side, - 1 _ Cx + D 2(1 -h«^"" 1+x^' 11 follows that C = and D = - 1/2, and we have, finally, 1 1 1 1__ ^.^. **--l 4('-l) 4(x+l) 2(x' + iy whieh gives J*_j^ = j,„g(i_|^_i ^arc tana;. V, §K>4] RATIONAL FLXCTIONS 211 Note. The preceding method, though absoluiely gMieral, it not always the simplest. The work may often be shorteoed by utinf a suitable device. Let us consideri for example, the integral /i dx (x«-l)- If n > 1, we may either break up the integrand into partial frac- tions by means of the roots + 1 and — 1, or we may uae a rednctioo formula similar to that for /.. But the most elegant method ia to make the substitution a: = (1 -f «)/(! — «), which gives J {x^ - 1)- 4- J «• ""• Developing (1 — «)*•-* by the binomial theorem, it only remaina to integrate terms of the form Azf^y where /& may be poeitire or negative. 104. Hermite's method. We have heretofore supposed tuai uie fraction to be integrated was broken up into partial fractions, which presumes a knowledge of the roots of the denominator. The»fol- lowing method, due to Hermite, enables us to find the algebraio part of the integral without knowing these roots, and it inrolTea only elementary operations, that is to say, additions, multiplications, and divisions of polynomials. Let f(x)/F{x) be the rational fraction which is to be integrated. We may assume that /(x) and F{x) are prime to each other, and we may suppose, according to the theory of equal roots, that Um polynomial F{r) is written in the form F(a-) = -Y,X}A';..-.Y;, where A'l, A,, • • •, A'^ are polynomials none of which have moltiplf roots and no two of which have any common factor. We may now break up the given fraction into partial fractions whoae tors are A'j, A J, • • •, A J : where .4, is a polynomial prime to A<. For, by the theory of high- est common divisor. '^^ v mwI i' are any two Txtlvnomiala which are J12 INDEFINITE INTEGRALS [V,§104 prime to each other, and Z any third polynomial, two other poly- nomials A and B may always be found such that BX-^-AY^^Z. Let us set .Y = Xi, K = A'J • • • JYJ, and Z =/(«). Then this identity becomes BX^^-AX\''Xl=^f{x), or, diriding by F(a;), It also follows from the preceding identity that if f{x) is prime to JP(x), A is prime to A'l and B is prime to Z| • • • AJ. Repeating the process upon the fraction B X\"'Xl and so on, we finally reach the form given above. It is therefore sufficient to show how to obtain the rational part of an integral of the form Adx f where ^(x) is a polynomial which is prime to its derivative. Then, by the theorem mentioned above, we can find two polynomials B and C such that B<f>(x)+C<t>Xx) = A, and hence the preceding integral may be written in the form If n is greater than unity, taking And integrating by parts, we get wheooa, substituting in the preceding equation, we find the formula /Ajg_ c r A,dx V,$104] RATIONAL FUNCTIONS 21S where ^^ is a new polynomial. If n > 2, we idaj apply Uia nm* process to the new integral, and ao on : the proceea may alwftyi be continued until the exponent of ^ in the denominator ia equal to one, and we shall then have an expression of the form /!?-«<-)-/'?■ where R{x) is a rational function of z, and ^ is a polynomial wboee degree we may always suppose to be less than that of ^, bat which is not necessarily prime to <^. To integratt* the latter form we must know the roots of <^, but the evaluation of tliis integral will intitK duce no new rational terms, for the decomposition of the fraction \l//<t> leads only to terms of the two types A Mx-^-N x-a (x-a)«-f/J«' each of which has an integral which is a transcendental function. This method enables us, in particular, to determine whether tho integral of a given rational function is itself a rational funetioD. The necessary and sufficient condition that this should be true is that each of the polynomials like ^ should vanish when the pi has been carried out as far as possible. It will be noticed that the method used in obtaining the for In is essentially only a special case of the preoeding consider the more general integral f, ^ AitO, B^^AC^O, (^x« + 2Bz + C7)» From the identity A{Ax* + 2Bx + C) - (ilx + B)« = AC' B* it is evident that we may write r dx ^ A r dz J (Ax* + 2J?x + C7)« AC'B*J {Aa* + IBs + Cy—« Integrating the last integral by parts, we find / Ax-^B . Am-k-B ^'^'''^^(Ax* + 2Bz + 0-^' nn-lHA^-^tBx-^Cr -I A r <» 214 INDEFINITE INTEGRALS [V, § 104 wlHlioe the preceding relation becomes / dz Az + B {Az^ + IBx + O" " 2(n - 1)(^C - B^){Ax^ + 2Bx + C)«-i 2fi - 3 A f dx ^2n-2 AC-B^J {Ax'^ + 2Bx + C^""! Continuing the same process, we are led eventually to the integral / dx Ax^ + 2Bx-\- C which is a logarithm if B^ - ^C>0, and an arctangent ilB^-AC<0. As another example, consider the integral f. 6x« + 3x - 1 , ax. (X8 + 3X + 1)8 From the identity 6x« + 3x - 1 = 6x(x2 + 1) - (x8 + 3x + 1) it is erident that we may write r_6x^+3x-l^^ r 6x(x^ + l) ^^_ r dx J (z« + 8x + 1)« J (x8 + 3x + 1)« J (x3 + 3x + l)^ Integrating the first integral on the right by parts, we find r 6(x« + i)dx ^ -X r dx J (z« + 3x + 1)» (x« + 3x + 1)2 J (x3 4- 3x + 1)^ whence the value of the given integral is seen to be 5x« + 8x - 1 / dx = — — — • (x» + 3x + 1)8 (x8 + 3x + 1)2 Note. In applying Hermite's method it becomes necessary to solve the fol- lowing problem : given three polynomials A, B, C, of degrees m, n, p, respectively^ two qf vthicK, A and B^ are prime to each other, find two other polynomials u and v miek tkat the reUUion Au + Bv = C is identically satisfied. In order to determine two polynomials u and v of the least possible degree which solve the problem, let us first suppose that p is at most equal to m + n — 1. Than w may take for u and v two polynomials of degrees n — 1 and w — 1, mptclivsly. The m-\- n unknown coeflBcients are then given by the system of m -f A linear non-homogeneous equations found by equating the coefficients. For the determinant of these equations cannot vanish, since, if it did, we could find two polynomials u and v of degrees n — 1 and m — 1 or less which satisfy tkt Idaotlty <Ati + Be = 0, and this can be true only when A and B have a II the da^rM of C Is equal to or greater than m + n, we may divide Chy AB tod obtain a remainder C whose degree is less than m-\-n. Ttun C = ABQ-\-C% and« making the aabetitutlon u - BQ = ui, the relation Au-\- Bv = C reduces to Aut -f Bf B C. This is a problum under th» first case. V,§103] RATIONA! M VCTI0N8 tl5 105. Integ:ral8 of the type /r(x, VA^ + SbT+C) dx. After th* integrals of rational functions it is natural to ooosider the inte- grals of irrational functions. We shall commenoe with the ease in which the integrand is a rational function of x and the iqaare tool of a polynomial of the second degree. In this case a simple tubititu- tion eliminates the radical and reduces the integral to the preceding case. This substitution is self-evident in case the expression under the radical is of the first degree, say ax -{- b. If we set ox -f- ^ v <*, the integral becomes j r{x, y/ax -f h)dx = f/^r and the integrand of the transformed integral is a rational ftmetioo. If the expression under the radical is of the second degree and has two real roots a and b, we may write and the substitution V X — a ^ Aa — bi* A 7 = tt or r = r-» x-b ' .1 - ^ actually removes the radical. If the expression under the radical sign has imaginary roots, the above process would introduce imaginaries. In o rder to get to the bottom of the matter, let y denote the radical V^x* -f 2Bx + C Then x and y are the coordinates of a point of the curve whose equation is (1) y« = Ax* + 2Bx -f- C, and it is evident that the whole problem amounts to expressing liie coordinates of a point upon a conic by means of rational functions of a parameter. It can be seen geometrically that this is possible. For, if a secant y-/5 = <(x-a) be drawn through any point (a, fi) on the conic, the coordinates of the second point of intersection of the secant with the oooic are given by equations of the first degree, and are therefore rational functions of t. If the trinomial Ax* + 2Bx + C has imaginary roots, the coefl- cient .4 must be positive, for if it is not, the trinomial will be negative for all real values of x. In this case the conic (1) is an 216 INDEFINITE INTEGRALS [V,^i06 hyperbola. A straight line parallel to one of the asymptotes of this hyperbola, y=zxy/A-\-t, enU the hyperbola in a point whose coordinates are C -t* r- C -f^ If A < 0, the conic is an ellipse, and the trinomial ^x^ + 2Bx + C must have two real roots a and b, or else the trinomial is negative for all real values of x. The change of variable given above is pre- cisely that which we should obtain by cutting this conic by the moTing secant y = t(x — a). As an example let us take the integral dx /; (x^ + k) Va;2 + k The auxiliary conic y* = a;* + A; is an hyperbola, and the straight line X + y = ^ which is parallel to one of the asymptotes, cuts the hyper- bola in a point whose coordinates are Making the substitution indicated by these equations, we find or, retoming to the variable x, /; dx X — -y/x^ + k X 1 (x« + A:)* A;Vx'» + A; k^/x^-^k k the right-hand side is determined save for a constant term In general, if ^4 C — B* is not zero, we have the formula dx 1 Ax + B {Ax* + 2Bx + C)* " AC -B* y/Ax^ + 25a; + C In tome enees it is easier to evaluate the integral directly without fMnoring the radical. Consider, for example, the integral dx /; / y/Ax* f 2Bx -f C V,ilOfi] RATIONAL FI'VCTIONS JIJ If the coetiicient A is positive, the integral may be writtflii r ^dx r VAdx J y/A^x* + 2ABx + AC J ^(Az-^B)*-^ AC ■- B** or setting Ax -^ B = t, -7= / . ^ : = "7=log(<-i-V<«^^C-B*). y/AJ ^t*^AC-B* WA ' Returning to the variable x, we havft the formula If the coefficient of x^ is negative, the integral may be written in the form C dx r y/ldx ^ J V- Ax^ + 25a; + C J y/ AC + B^ -(Ax - B)** The quantity AC + B'^ is necessarily positive. Henoe, twaUng ^ii^ substitution Ax- B = t VTcwTb*, the given integral becomes \ r dt 1 — = I ■ • = —7= arc sin ^. Hence the formula in this case is r dx 1 . Ax-B I ^ = — ^ arc sin . =• J V- ^x» + 2fia; + C Va >MC + B« It is easy to show that the argument of the arcsine Tariee from — 1 to + 1 as X varies between the two roots of the trinomial. In the intermediate case when A ssO and B ^ 0, the integral i« algebraic : / dx 1 V2Bx -f C ^ Integrals of the type dx = - V2Bx + C. / (x - a)V.4x« + 2J5X + C 218 INDEFINITE INTEGRALS [V, §100 reduce to the preceding type by means of the substitution x = a-\-l/y We find, in fact, the formula r dx ^ _ r dy J {x-a) V^a5« + 2Bx -f- C J \/ A^'/ -\-2B-^y -^ C^ where i4i = ^a« -f iBa + C, B^z=Aa-\-By Ci = A. It should be noticed that this integral is algebraic if and only if the quantity a is a root of the trinomial under the radical. Let us now consider the integrals of the type /Va;^ -f A dx. Inte- grating by parts, we find f^:^'+Adx = xV^^^TA- f-^£g J J vvT dx . A On the other hand we have f^^=fv^^T^dx-f-^ 7 V^^Ta J J Var^-f = / Vx^ H- Adx - A log(a: + Vx^-f-vl). From these two relations it is easy to obtain the formulae (2) j^x* + Adx= ^v^M^+|log(xH-V^M^), ^^^ /^Sf^ = |ViM:^-|iog(. + V^rr7^. The following formulsB may be derived in like manner : (4) Jv^5Tr^d»= fV^F3p+^ . X arc sin -> a . X arc sin — a Af !jI!* ^'^ ** ^ kyperboU. The preceding integrals occur in the evaluation Of tlM MM of a MCtor of an ellipse or an hyperbola. Lot us consider, for Um hyperbola V, § 108] UATIOXAI, FI'VCTIONS 219 and let uh try to flii<l ii .- u . ;i oi a gegmeut ^IfP boiUMl«d bjr Um m« AM, tte X axis, aud the urdiuuic MI'. Tbk Area !■ eqiul to th* <**< yfi iit Inuaml X '6 V3?^^<fe, that is, by the formula (2), i^[,V^--..o.(i±v^] But MP = V = (Va) Vz> - a*, and the term {b/2a)x V«*-^ to pradMly Um area of the triangle OAfP. Hence the area 5 of the aeetor 0AM, boondtd bj the arc AM and the radii vectores OA and OM, is This formula enables us to exproH the coordinates x and y of a point M of the hyperbola in terms of the area S. In fact, from t^e above and from the equation of the hyperbola, it is easy to show that a b Pio.» e"i*, or The functions which occur on the right-hand side are called the cosine and 'sine : coshx e* + € ih« = ff'-e-J 2 S The above equations may therefore be written in the form 23 X = a cosh — » ab y = 6 eiDh Theee hyperbolic functions possees properties analofona to Ihoee of the trffo> nometric functions.* It is easy to deduce, for Inetaiico, the foUowiag cosh'z - sinh*x = 1, cosh (z H- y) = cosh x coah y + sinh z dnh y, sinh (x + y) = sinh z cosh y •)- sinh y ooehz. • A table of the I<>)n»rithm8 of theee funetioae for poelttTe Taloee of the is to be found iu Houel's Recueii du/ormuiM nmmtriqw$». sso INDEFINITE INTEGRALS [V,§107 bs ibown in like manner that the coordinates of a point on an ellipse ■ij be exprwMed In termi of the area of the corresponding sector, as follows : 28 aco8— •» ab . . 25 ftsin— -. db la the OM* of a circle of unit radius, and in the case of an equilateral hyperbola whoM tf*"***^ is one, these formuls become, respectively, 2 = C06 25, y = sin25; z = coah25, y = sinh2iS. It li erident that the hyperbolic functions bear the same relations to the equi- lateral hyperbola as do the trigonometric functions to the circle. 107. RectUcatioB of the parabola. Let us try to find the length of the arc of A parabola 2py = x* between the vertex O and any point M. The general fonnula giTW arc »'-xv-(s'-r^- or, applying the formula (2), arc OM X Vz« + pg 2^ |,o.(-^fI^ aA Tbe algebraic term in this result is precisely the length MT of the tangent, for we know that 0T=: z/2, and hence V** ^^*1 ,,^ .. . X* X* . X2 X2(x2 + p2) wn *^ 4 4pa 4 4p2 If we draw the straight line connecting T to the focus F, the angle MTF will be a right angle. Hence we have FT =v? 2 X2 ] / + - = Wp2-|.x2, r r' Fio.23 ll UDgMt at JT' to the X axis. OM' = arc OM. poiiUoo r such that If'r = MT, and the focus by laying off T'F' = TF on a line parallel to the y a^is, X and r of the point F' are then whence we may deduce a curi- ous property of the parabola. Suppose that the parabola rolls without slipping on the x axis, and let us try to find the locus of the focus, which is sup- posed rigidly connected to the parabola. When the parabola The point T has come into a F is at a point F' which is The coijrdi- X«aroOJf-jrr = |log(?±2^^±Z), V,5108] RATIONAL FUNCTIONS and the equation of the locus ia giTen by elimlnaUog x bt t i ma Umm two tioiiH. Fmiii iht* firHt w»< find to which we may add the equation _H X - Vx* + p« = - pe »• aince the product of the two left-hand aides la eqiuU to - ^. SabCncUiif two equations, we iind and the desired equation of the locus ia IX t.T' -f(---'-)=l«-T This curve, which is called the catenary^ la quite 9Uj to oooatmet. Ita fi is somewhat similar to that of the parabola. 108. Unicursal curves. Let us now consider, in geoenly the inte- grals of algebraic functions. Let (6) F(x,y) = be tlie equation of an algebraic curve, and let R(iy y) be a rational function of x and y. If we suppose y replaced by one of the roota of the equation (6) in H(x, y), the result is a function of the single variable x, and the integral / R(Xy y)dx is called an Abelian integral with respect to the cnrre (6). When the given curve and the function R(Xf y) are arbitrary these inte- grals are transcendental functions. But in the particular case where the curve is unicursal, i.e. when the cxxirdinates of a point on the curve can be expressed as rational functions of a variable param- eter t, the Abelian integrals attached to the curre can be reduoed at once to integrals of rational functions. For, let be the equations of the curve in terms of the parameter t Taking t as the new independent variable, the integral becomes fR(x, 1/) dx =Jr lJ[t), ^(t)']f{t)di, and the new integrana is evidently ritiiollsL 2SS INDEFINITE INTEGRALS [V,§108 It is shown in treatises on Analytic Geometry * that every uni- ennai curve of degree n has (n — l)(n - 2)/ 2 double points, and, ooQTersely, that every curve of degree n which has this number of doable points is unicursal. I shall merely recall the process for obtaining the expressions for the coordinates in terms of the param- eter. Given a curve C. of degree n, which has S = {n — l)(n — 2)/2 double points, let us pass a one-parameter family of curves of degree •I — 2 through these S double points and through n — 3 ordinary points on C,. These points actually determine such a family, for (n-l)Cn-2) , ^^ 3_ (n-2)(n+l) ^^ 2 2 whereas (n — 2)(n -f l)/2 points are necessary to determine uniquely a curve of order n — 2. Let P(x, y) -f tQ{Xy y) = 0\)Q the equation of this family, where ^ is an arbitrary parameter. Each curve of the family meets the curve C, in n(n — 2) points, of which a certain num- ber are independent of t, namely the n — 3 ordinary points chosen above and the 8 double points, each of which counts as two points of intersection. But we have n - 3 -f 28 = n - 3 -f (w - l)(7i - 2) = 7i(7i - 2) - 1 , and there remains just one point of intersection which varies with t. The coordinates of this point are the solutions of certain linear equa- tions whose coeflBcients are integral polynomials in t, and hence they are themselves rational functions of t. Instead of the preceding we might have employed a family of curves of degree n — 1 through the (n — l)(n — 2)/2 double points and 2n — 3 ordinary points chosen at pleasure on C.. If n s= 2, (n — l)(n — 2)/2 = 0, — every curve of the second degree is therefore unicursal, as we have seen above. If w = 3, (» — l)(n — 2)/2 = 1, — the unicursal curves of the third degree are those which have one double point. Taking the double point M origin, the equation of the cubic is of the form <^«(^»y)4-</»,(a;, 2/) = 0, where ^, and ^ are homogeneous polynomials of the degree of their indioee. A secant y = tx through the double point meets the cubic in a single variable point whose coordinates are ^•a»0 "^ Mht) •Stt, •.§., NI«WMflowild, Court de Q4omitH« ancUytiqm., Vol. II, pp. 9&-114. V,fl08] UATIONAL FUNCTIONS Sf| A unicursal curve of the fourth degree hai three doable points. In order to find the coc^rdinates of a point on it, we ihookl paee a family of conies through the three double points and through anoUMr point chosen at pleasure on the curve. Every conic of this familj would meet the quartic in just one point which Taries with the parameter. The e(|uation which gives the sbeeissaD of the points of intersection, for instance, would reduce to an equation of the first degree when the factors corresponding to the doable points hsd been removed, and would give x as a rational fnnetion of the parameter. We should proceed to find y in a similar manner. As an example let us consider the lemniscate (aj« + y«)« = «'(x»-/), which has a double point at the origin and two others at the imagi- nary circular points. A circle through the origin tangent to one of the branches of the lemniscate, x« + y» = f(x-y), meets the curve in a single variable point. Combining these two equations, we find or, dividing by x — y, This last equation represents a straight line through the origin which cuts the circle in a point not the origin, whose coordinates ars *- t'j^a' ^" <* + a« These results may be obtained more easily by the following process, which is at once applicable to any unicursal curve of the fourth degree one of whose double points is known. The secant y — Xz cuts the lemniscate in two points whose coordinates are X = 14- X« y-X«. The expression under the radical is of the second degree. Hsne^ by § 105, the substitution (1 - X)/(l + X) - (a/f)« removes the radi- cal. It is easy to show that this substitution leads to the just found. 224 INDEFINITE INTEGRALS [V,§109 NoU, When a plane curve has singular points of higher order, it oaa be shown "tfcat each of them is equivalent to a certain number of i^^l^i^ double points. In order that a curve be unicursal, it is suffi- eient that its singular points should be equivalent to {n — l){n — 2)/2 itolated double points. For example, a curve of order n which has a multiple point of order n — 1 is unicursal, for a secant through the multiple point meets the curve in only one variable point. 109. Integrals of binomial differentials. Among the other integrals in which the radicals can be removed may be mentioned the follow- ing types: / r\x, {ax + hY\dx, I R{x, -s/ax -f- h, Vex -\-d)dXy R(x'j x"', x*", ■•)dx, I' where R denotes a rational function and where the exponents a, a\ a"f ••• are commensurable numbers. For the first type it is sufficient to set aa; -f i = ^. In the second type the substitution €uc -\- b = t* leaves merely a square root of an expression of the •eoond degree, which can then be removed by a second substitution. Finally, in the third type we may set x = t^, where D is a common denominator of the fractions a, a', a", • • •. In connection with the third type we may consider a class of di£Ferentials of the form x''(aaf' + bydxj which are called binomial differentials. Let us suppose that the three exponents w, n, p are commensurable. If jt> is an integer, the expression may be made rational by means of the substitution ' = ^» ae we have just seen. In order to discover further cases of integrability, let us try the substitution ax"" + b = t. This gives The transformed integral is of the same form as the original, and the exponent which takes the place of p is (m + l)/n - 1. Hence the integration can be performed if {m + l)/n is an integer. V,§109J BATIONAL FUNCTIONS On the other hand, the integral may be writteo in the form f- whence it is clear that another case of integrabilitj is that in whieh (w + np -f l)/n = (m + l)/n -f p is an integer. To sum up, tlM integration can be performed whenever one of the three nmmhere Pj (m -f l)/n, (m + l)//i -^-p is an integer. In no other case eaa the integral be expressed by means of a finite number of elementarj functional symbols when m, n, and p are rational. In these cases it is convenient to reduce the integral to a simpler form in which only two exponents occur. Setting a«* a bif we find ' l/6\-i-i x = l- r, cfa = Mf <• dt, n \a x'^(ax''-{-bydx = -l-] "It* 'f^tydi. Neglecting the constant factor and setting g = (m + l)/n — 1, we are led to the integral / t^(i'^tydt. The cases of integrability are those in which one of the three m bers p, q, p -\- q w an integer. If /) is an integer and q = r/«, we should set t = t/*. If y is an integer and p = r/«, we should set 1 4- < = u\ Finally, if jd -f- y is an integer, the integral may be written in the form /'"•(¥)'-. and the substitution l-{- f= Ui'y where p = r/#, remores the radical As an example consider the integral / xy/l-k-T^dx. Here m = 1, n = 3, ;> = 1/3, and {m + V)/n +p - 1. Heooe thii is an integrable case. Setting x* = r, the integral b eoo nm and a second substitution 1 4- f = /u* remoTes the r adio al. INDEFINITE INTEGRALS [V,§110 II. ELLIPTIC AND HYPERELLIPTIC INTEGRALS UO. Reduction of integrals. Let P{x) be an integral polynomial of degree p which is prime to its derivative. The integral JR[x,Vp{^)]dx, where R denotes a rational function of x and the radical y = -\/P(x), eannot be expressed in terms of elementary functions, in general, when p is greater than 2. Such integrals, which are particular etMt of general Abelian integrals, can be split up into portions which rwalt in algebraic and logarithmic functions and a certain number of other integrals which give rise to new transcendental functions which cannot be expressed by means of a finite number of elemen- tary functional symbols. We proceed to consider this reduction. The rational function /2(x, y) is the quotient of two integral polynomials in x and y. Replacing any even power of y, such as y**» by [.^{^yfi and any odd power, such as y^' + S by y [P(«)]S we may evidently suppose the numerator and denominator of this frac- tion to be of the first degree in y, -,, . A-\- By R(x, y)= — ■ — ^, where Aj B, Cy D are integral polynomials in x. Multiplying the numerator and the denominator each by C — Dy, and replacing y^ by P(ar), we may write this in the form ^(*. y) = —j^> where F, G, and K are polynomials. The integral is now broken up into two parte, of which the first Jf/K dx is the integral of a rationil function. For this reason we shall consider only the second integral fOy/K dx, which may also be written in the form /; Mdx n-Vp(x) where If and AT are integral polynomials in x. The rational frac- tion M/N may be decomposed into an integral part E(x) and a ram of partial fractious V, §ii()j hhhirilL ASU llll'tKKLUPTIC INTKliRALS 227 where eai?h of the polynomials A\ is prime to its derivative. We shall therefore have to consider two types of integrali, ^x) J A'-V7xi) If the degree of P(x) is p^ all the integraU K. may he exprtsied in terms of the first p - 1 of them, Ko, K,, •••, K^.,, and certmim algebraic expressions. For, let us write P(x) = aoJB' H- ajaj"-* + • • •• It follows that :^ (x- vTy^) = mar- - » Vi^) -H i^^^^^l ^ 2wjr— »P(g) 4- afP*(a?) 2 VP(x) The numerator of this expression is of degree m +/' — 1, and its highest term is (2m -\- p)anX'*-*-p-K Integrating both sides of the above equation, we find 2x'-VP{x)=(2m-^p)a,r^^,.^ + .-•, where the terms not written down contain integrals of the type y whose indices are less than m -{- p — 1. Setting m = 0, 1, 2, • • •, successively, we can calculate the integrals K,_i, K,, ••• suooes- sively in terms of algebraic expressions and the /> — 1 integralf With respect to the integrals of the second type we shall distin- guish the two cases where A' is or is not prime to P(x). 1) If X is prime to P(x), the integral Z^ reduces to the sum of an algebraic term, a number of integrals of the type K*, and a integral Bdx /: XWP(x) where B is a polynomial whose degree is less thorn that of X. Since -Y is prime to its derivative X' and also to P(ap), JT* is prims to PX'. Hence two polynomials X and ft can be found such that XX^ -^ fiX'P = A, and the integral in question bfsaks up into two parts: J A-\ /\x) J \/P{x) J INDEFINITE INTEGRALS [V, §110 The first part is a sum of integrals of the type Y. In the second integral, when » > 1, let us integrate by parts, taking which gives r ^Vpx'dx ^ -mVp ^ 1 CML±JtELdx J X* (n-l)A"-' n-lJ 2X^-'Vp(^ The new integral obtained is of the same form as the first, except ihftt the exponent of X is diminished by one. Repeating this prooees as often as possible, i.e. as long as the exponent of X is greater than unity, we finally obtain a result of the form / Adx r Bdx r cdx dVp X*y/P(x) J aVp J Vp Z" where B, C, D are all polynomials, and where the degree of B may always be supposed to be less than that of X. 2) If X and P have a common divisor Z), we shall have X — YD, P ra SD, where the polynomials D, S, and Y are all prime to each other. Hence two polynomials X and fi may be found such that A = X/)* -I- fiY", and the integral may be written in the form r Adx ^ r Xdx Tju J x*Vp J y«Vp J z)" dx Vp The first of the new integrals is of the type just considered. The 9 9oemd i$UegrcLly /fjL dx ly'Vp whmrt D is a factor of P, reduces to the sum of an algebraic term and a number of integrals of the type Y. For, since />• is prime to the product D'S, we can find two poly- BomiaU Aj and fi^ such that \,JJ^ -f fi^D'S = fi. Hence we may write J iryTp J y/p^J d^Vp"^^' Replaeing P by DS, let us write the second of these integrals in the V.§110] ELLIPTIC AND HYPERELLIPllt JMLcltM > 229 and then integrate it by parts, taking which gives J L^y/P J V? (n - !)/>•-». 2n-l J ir^VP This is again a reduction formula ; but in this case, since the expo- nent n — 1/2 is fractional, the reduction may be performed erao when D occurs only to the first power in the denominator, ^^ we finally obtain an expression of the form dx kVp J r^y/p />• J V? ' where H and K are polynomials. To sum up our results, we see that the integral / Mdx nVp can always be reduced to a sum of algebraic terms and a number of integrals of the two types J V^' J X^ dx Vp where m is less than or equal to /> — 2, where A' is prune to lU derivativ.e A" and also to P, and where the degree of A'l is lest thao that of X. This reduction involves only the operations of aAlifiii, multiplication, ami division of polynomials. If the roots of the equation A' = are known, each of the rational fractions Ai/X can be broken up into a sum of partial fractioni of the two forms A Bx-^C x — a (« — a)* + /8** where i4, B, and C are constants. This leads to the two new tjpee C)<£ar r dx r (Bx -f J (x - a)VP(^' J [(x - ay + which reduce to a single type, namely the first of theee, if we to allow a to have imaginary values. Integrals of this tort ere 280 INDEFINITE INTEGRALS [V, §110 ealled imUgraU of the third kind. Integrals of the type F„, are eiUed vUegmU of the first kind when m is less than p/2 — 1, and are called integrals of the second kind when m is equal to or greater than p/2 — 1. Integrals of the first kind have a characteristic property, — they remain finite when the upper limit increases indefinitely, and also when the upper limit is a root of P(x) (SS S^» dO); but the essential distinction between the integrals of the leoond and third kinds must be accepted provisionally at this time without proof. The real distinction between them will be pointed out later. Note. Up to the present we have made no assumption about the degree p of the polynomial P{x). If p is an odd number, it may always be increased by unity. For, suppose that P(x) is a poly- nomial of degree 2q —1: P(x) = ^oa^'-' + A.x^''-' -}-... + A^_,. Then let us set a; = a -f- 1/y, where a is not a root of P(x). This giTes where Pj (y) is a polynomial of degree 2q. Hence we have and any integral of a rational function of x and VP(ar) is trans- formed into an integral of a rational function of y and VW^). ConTersely, if the degree of the polynomial P{x) under the radi- cal U an even number 2y, it may be reduced by unity provided a root of P(x) is knoum. For, if a is a root of P(x), let us set « » a + 1/y. This gives P(x) = P'(«)- + ... + ^^^^J- = ^lIl^, y (2q)l y'" y*« ' whara />|(y) U of degree 2y - 1, and we shall have a«ca the integrand oQhe transformed integral will contain no dher radical tlian V/»i(y). V, §111] ELLIPTIC AND IIYPKRKI.IJPTir TXTFORaI^ 281 111. Case of integration m algebraic termi. We Mve jun Men UxMl An UUtgitl of the form fRl X. v7^i)]dx can always be reduced by mewaa of elementary openitloM to the iOB <d to IbI»- gral of a rational fraction, an algebraic expreeiion of tbe form O VP^)/L, tad a number of integrals of the first, second, and third kinds. Since we can also find by elementary operations the rational part of the integral of a ratiooal fraction, it is evident that the given integral can always be reduced to tbe fom J/(x, vm]dx = Fix, vF(ij] + r. where F is a rational function of x and VP(x), and where 7 is a sum of inl^ grals of the three kinds and an integral fXi /Xdx, X being prime to tie deriva- tive and of higher degree than .Yi. Liouville showed that if the given integrai is integrable in algebraic terms, it is equal to F[Xf VP(x)]. We dionld tlMre> fore have, identically, «[x, VP(i)] = I Kx, V^)]{. and hence 7=0. Hence toe can discover by means of multiplicationM and difMonM <ifpolifmamkU» whether a given integral is integrable in aigebraic terms or noC, ami in earn U it, the same process gives the value of the integral. 112. Elliptic integrals. If the polynomial P{x) is of the seoond degree, the integration of a rational function of x and P{x) eao be reduced, by the general process just studied, to the calculation of the integrals r dx r dx J Vp(x) ' J {x- a)y/P{x) which we know how to evaluate directly (§ 105). The next simplest case is that of elliptic integrals, for which l\x) is of the third or fourth degree. Either of these oases can be reduced to the other, as we have seen just above. Let P{x) be a polynomial of the fourth degree whose coefficients are all real and whose linear factors are all distinct. We proceed to show that a real substitution can always be found which carries P{x) into a polynomial each of whose terms is of even degree. I^t a, b, c, d be the four roots of P{x). Then there eiists an involutory relation of the form (7) Ix'«" + Af («' + «") + JV == 0, INDEFINITE INTEGRALS [^,§112 which in satisfied by or' = a, x" = ft, and by x' = c, a;" = d. For the OOiAoients L, A/, ^' ueed merely satisfy the two relations /:a^ + jjf(a + ft)-f iV' = 0, which ai« eridently satisfied if we take Ltma-^b^e — df M ^cd — abf N = ab(c -\- d) — cd(a -\- b). Let a and ^ be the two double points of this involution, i.e. the rooto of the equation Lu* + 2Afu + N = 0. These roots will both be real if (cd - aby-(a + b - c - d)[ab(c + d)- cd(a + ft)] > 0, thatia,if (8) (a -c)(a- d)(b - c)(b -d)>0. The roots of P{x) can always be arranged in such a way that this condition is satisfied. If all four roots are real, we need merely ohc¥)ee a and ft as the two largest. Then each factor in (8) is positive. If only two of the roots are real, we should choose a and ft as the real roots, and c and d as the two conjugate imaginary roots. Then the two factors a — c and a — d are conjugate imaginary, and so are th^ other two, ft — c and b — d. Finally, if all four roots are imaginary, we may take a and ft as one pair and c and d as the other pair of conjugate imaginary roots. In this case also the factors in (8) are conjugate imaginary by pairs. It should also be noticed that these methodB of selection make the corresponding values of i, M, N real The equation (7) may now be written in the form (9) ^^^l^-f ^'-^=0 ^^ x^-fi^ X--P \' If we Mt (« - a)/{x _ ;3) = y, or a; = {Py - a)/iy - 1), we find Pi(y) i* a new polynomial of the fourth degree with real whose roots are g -• g ft — g e— a d — a a-^fi' b-p* e-p' JZTp II if eridnt from (9) that these four roots satisfy the equation V,J112] ELLIPTIC AND HYPEKELLIPTIC INTEGRALS «tt y' 4- y" = by pairs ; henoe the polynomUl Px(}f) oonUini do ierm of odd degree. If the four roots a, by c, d satisfy the equation a-^htse-^d^wt shall have Z. = 0, and one of the double points of the inrolutioD lie* at infinity. Setting a = — N/2My the equation (7) takat the form rr' — a + as" — a = 0, and we need merely set x = a -f y in order to obtain a polynomial which contains no term of odd degree. We may therefore suppose P(x) reduced to the ^nffnii?a1 form P(x) = ^o«* + iijX* + At. It follows that any elliptic integral, neglecting an algebraio term and an integral of a rational function, may be reduced to the ■am of integrals of the forms /dx r xdx r V%x*T^47xH^' J V^o«*+^iaf*+^«' J V^ and integrals of the form dx !-. (x - a) \MoX* + ^iX« -f -4, The integral dx •■=r- is the elliptic integral of the first kind. If we consider Xy on the other liand, as a fuuction of t/, this inverse function is called an elliptic function. The second of the above integrals reduces to an elementary integral by means of the substitution ac* = ti. The third integral r ^dx J V.4o«*-|-i4iX«-f i4, is Legendre's integral of the second kind. Finally, we have tba identity r d^ =r '^^ +af ^ ^ . J (X - a) Vp(i) J (x« - a«)>/P(x) •/(«•- a«) VP(x) The integral r dx J (x« + A)Vi4pX* + i4iX« + ^, is Legendre's integral of the third kind. {(4 INDEFINITE INTEGRALS [V, § ns TheM eUiptic integrals were bo named because they were first met with in tl.e problem of rectifying the ellipse. Let x = aco8^, 7/ = b8m<l> be the codidinates of a point of an ellipse. Then we shaU have ds»^dx^^dy'-(a^ sin^</» + b^ cos^*^) d<l>\ or, setting a* — A* = c*a*, ds = aVl — e'cos*<^ dtf*- Hence the integral which gives an arc of the ellipse, after the sub- stitution cos ^ = ^ takes the form s= a I — , dt = a I ' , „^ ^ ==dt. It follows that the arc of an ellipse is equal to the sum of an inte- gral of the first kind and an integral of the second kind. Again, consider the lemniscate defined by the equations An easy calculation gives the element of length in the form d^^d^^-dy"^ -^-—l dt\ t ~\~ CL Hence the arc of the lemniscate is given by an elliptic integral of the first kind.* lit. PMado-«lUptic integrals. It sometimes happens that an integral of the form /F[x, VP(x)]dx, where P(x) is a polynomial of the third or fourth degree, can be expressed in terms of algebraic functions and a sum of a finite of logarithms of algebraic functions. Such integrals are called pseudo- Thie hftppens in the following general case. Let (10) Lx'x" + M{x' + X") + iV = b$ on hnohiiory relation which eatablishea a correspondence between two pairs of tktfomr rooU of the qttartic equation P(x) = 0. If the function f{z) be such that tktrdaUtm i§ idtntkaUy ioti^fied, the inUffral /[/(x)/ VP(x)] dx is pseudo-eUiptic. * TlUi to e eommoD property of a whole class of curves discovered by Serret (Onm 4f C'o/cu/ diffirentUl 9t integral, Vol. II, p. 264). V.JlKiJ ELLIPTIC A.NiJ ail-hlUuLLil'llL IMtCiUAI^ Let a and /9 be thu double puinu of the iDTOlalloil. Am w« tep seen, the equation (lOt may be written in the fonn (12) ?1Z^ + 51ZJU0. Let u« now make the lubetitution (x — a)/{x - fi)sy. This fl?« " (l-V)' " (!-»)♦' and consequently dg _ {a — fi)dy where Pi (y) is a polynomial of the fourth degree which contain* no odd powen of y (§ 112). On the other hand, the rational fraction /(z) goes over into a rational fraction <p{y), which satisfies the identity ^{y) + ^<- y) = 0. For If two values of x correspond by means of (12), they are transformed Into two values of y, say y' and y", which satisfy the equation f' -¥ u" = 0. It iaoridaBt that <p{y) is of the form y^{y^), where ^ is a rational function of y*. the integral under discussion takes the form / y^iy*)dy y/AQy*-^Air/* + At and we need merely set y* = zin order to reduce it to an elementary integral. Thus the proposition is proved, and it merely remains actually to carry out the reduction. The theorem remains true when the polynomial P{x) is of the third degree, provided that we think of one of its roots as infinite. The d ei B O M U atkm is exactly similar to the preceding. If, for example, the equation P{x) = is a reciprocal equation, one of the involutory relations which interchanges the roots hy pairs is x'z^ = 1. Henee, if /(x) bo a rational funct ion which satisfies the relation /{x) + /(1/x) = 0, the integral /[/(x)/VP(x)] dx is peeudo^lliptic, and the two subsUiulioos (x — l)/(x -f 1) = y, y2 = z, performed In order, transform it into an elementary integral. Again, suppose that P(x) is a polynomial of the third P(«) = x(x ■,(.-i) Let us set a = 00, 6 = 0, c = 1, d = l/lfi. There exist three Involutory rela- tions which interchange these roots by pairs : fc«x" *«(i-o i-*«jr Hence, if /(x) be a rational function which satisilee one of the U 986 INDEFINITE INTEGRALS [V,§114 f{x)dx f Vx{l-x){l-k^x) Is pModfKeUipUc. From this others may be derived. For instance, if we set s = I*, the pnoeding integral becomes / 2f{z*)dz y/{l-z^){l-k^z^) whenea it follows that this new integral is also pseudo-elliptic if f{z^ satisfies OM ol U»e identities The flrat of these cases was noticed by Euler.* III. INTEGRATION OF TRANSCENDENTAL FUNCTIONS 114. Integration of rational functions of sin x and cos x. It is well known that sinx and cosx may be expressed rationally in terms of tan x/2 = t. Hence this change of variable reduces an integral of the form / i? (sin ic, cosa;)<^a; to the integral of a rational function of t. For we have « = Zarctan^ (fo = j-— , sina; = j— ^, cosx = ^— -^> and the given integral becomes where ♦(^) is a rational function. For example, r dx cdt , -. — = log tan jiA / Binx • Hoe II«rmit«'s lithographed Courn, 4th ed., pp. 25-28. V,§114] TRANSCENDENTAL FUNCTIONS S|7 The integral / [l/cos x] dx reduces to the preceding bj mMni of the substitution x = 7r/2 — y, which gives /^=-'o«-(M)='<.-(M)- The preceding method has the advantage of ganendity, hot it is often possible to find a simpler substitution which is eqnallj suc- cessful. Thus, if the function /(sin a;, cos x) has the period w, it is a rational function of tan a;, F(tan x). The substitution tan x s i therefore reduces the integral to the form /,(.«. .,^./.!m'. As an example let us consider the integral dx I: A cos* a; + B sin a; cos X + C sin* x + />* where A, B, C, D are any constants. The integrand evidently hsM the period ir ; and, setting tan x = ^, we find t . . C Hence the given integral becomes iT?' smxcosx = j-j-y,, sin«x=:j-^ /: dt The form of the result will depend upon the nature of the of the denominator. Taking certain three of the ooeffieients sero^ we find the formulae /dx C dx — ^ = tanx, \- =logtanx, cos'x J sin X COS X /; dx = — cot X. sin"x When the integrand is of the form R(%\tl x) cos x, or of the form /^(cosx) sinx, the proper change of variable is apparent In the first case we should set sin x = ^ ; in the second case, ooec ■* I. It is sometimes advantageous to make a first substtliitioo in older to simplify the integral before proceeding with the geDeral method. For example, let us consider the integral / dx acosx4-6 8inx + e 2S8 INDEFINITE INTEGRALS [V, § 114 whew a, 6, e are any three constants. If p is a positive number and ^ an angle determined by the equations we shall have and the given integral may be written in the form / dx _ r dy pco8(x-<^)+c J p cosy -he' where x — ^ = y. Let us now apply the general method, setting tan y/2 = t. Then the integral becomes 2dt f. P + c + (c- p)t^ and the rest of the calculation presents no difficulty. Two different forms will be found for the result, according as p^ — c^ = a^ -\-b^— c^ 18 positive or negative. The integral " m cos X -{- n sin x +p j ; — : ax a cos X -{- sm x -\- c f may be reduced to the preceding. Eor, let u = a cos x -\- h sin x -\- c, and let us determine three constants X, p., and v such that the equation du m cos X + n sin x-\-p = \u-{-p.- — hv it identically satisfied. The equations which determine these num- bers are m = Xa + pjbf n = \b — pa, p = \c -{- v, the first two of which determine X and p.. The three constants hav- ing been selected in this way, the given integral may be written in the form / du dx = Xx -\- pilog u -h V I «* J a cos X H- i sin X -H c Let OB try to evaluate the definite integral dx £ l-)-fOOS» « where |e|<l. vr,§iifi] TRANSCENDENTAL FUNCTIONS 2S9 Cooiidering it llnl •■ an indefinite integnd, we find ■nocuwlnlj f ^ ^2C ^ -r > C ** Jl + 6ooe« Ji + e + (l-«)<« Vl - > J 1 + 1^' by means of the sttoceeeiye subetitutione tanx/SsC, (ss«V(l-|- e)/(i > i^ Hence the indefinite integrml ia eqoal to vT-^ arc tan m-t) As z varies from to ;r, V(l - e)/(l + e) tan x/2 ineraMet from to 4* •, aad the arctangent varies from to ie/2. Hence the gifMi <*^*^««tt liiiif,nl Is afMl to 7r/V(l - c'-O- 115. Reduction formulae. There are also certain classes of intograls for which reduction formulaB exist. For instaooey the fonnnla lor the derivative of taii*"*x may be written d_ dx whence we find — (tan"->x) = (n -1) tan-«a5(l+ tan«x), /tan«-»x r , ^ tan*x<w;= I tan"""a;ttar. The exponent of tan x in the integrand is diminished by two units. Kepeatecl applications of this formula lead to one or the other of the two integrals jdx = x, I tanxr/xss— logcosx. The analogous formula for integrals of the type /cot" xdlx is /cot"~*j; r coV'xdx = I cot"""x<ix. In general, consider the integral 8in"a;co8"a;<ix, /■ where m and n are any positive or negative integers. Wheo one of these integers is odd it is best to use the change of vmriable ghreo above. If, for instance, n = 2;) -f 1, we should set sinx s (, whieh reduces the integral to the form /<"(!— <*)'<ft. Let us, therefore, restrict ourselves to the case where m and n are both even, that is, to integrals of the type f ^=s I sin'^xooshadb, 240 INDEFINITE INTEGRALS CV,§116 which may be written in the form / = / sin'^'^ascos^'ajsincccfe. Taking oo«»"x sinxdx as the differential of [- l/(2n 4-l)]cos2»+ia;, an integration by parts gives /^,= -riB--.^ + ^/sin--xcos-x(l-8in».)<i., which may be written in the form _ 8in**'"^agco8*'"'"^g 2m— 1 j (-^J i-.. ""■" 2(m + w) 2(m + w) ^«-i.«* This fonnula enables us to diminish the exponent m without alter- ing the second exponent. If m is negative, an analogous formula may be obtained by solving the equation (A) with respect to I^a-x^n and replacing m by 1 — m : ^^ -'—.."" l-2m "^ l-2m ^i-«,n' The following analogous formulae, which are easily derived, enable U8 to reduce the exponent of cos x \ J __ sin*"'-*-^xcos'"~^a; 2n—l j ^' -*".• " 2(m-fn) 2(m + w) ^m.— i' /D^ /" - - «^'^*"'"^^^<^Q8^"'*^ , 2(m + l-n) ^ ^^' ^-.-." i_2» "^ i-2n ^-.-n+r Repeated applications of these formulae reduce each of the num- bers m and n to zero. The only case in which we should be unable to proceed is that in which we obtain an integral /„ ,j, where m + n = 0. But such an integral is of one of the types for which reduction for- mula were derived at the beginning of this article. lit. WalUt' formnUB. There exiat reduction formulas whether the exponents m and n are •reo or odd. Am an example let lu try to eyaluate the definite integral w /»= r*8ln«"zda;, whM« m U a poaltlTe integer. An integration by parts gives J aln— »«iin«d«»-[oos»sin"-»x]» + (m-l) r'8in"-5jCos«xda;, V,$117J TRANSCENDENTAL FUNCTIONS S41 whence, noting that coex sin^-^z Taniahat at boib »■«<«#, wa fln4 tha fooNlA w /« = (m - 1) pain— »x(l - ^n*x)dx = (m - !)(/— t - /^, which leads to the recurrent formula (13) /, = !!^^7,_,. m Repeated applications of this formula reduce the given integral to /© = r/a if m is even, or to Ii = \ if m ia odd. In the former caM, taking m - 2p aiid replacing m succeasively by 2, 4, 6, • • • , 2p, we find It --Jo, A = -/i, , ^«p= -J— ^S,-lf or, multiplying these equations together, 1.8.6...(2p-l) jr '^ 2 . 4 . 6 . . • 2p 2 Similarly, we find the formula 2 . 4 . 6 . . . 2p 2p + l 1 . 3 . 6 . • (2p + 1) A curious result due to Waliis may be deduced from these formulc It U evident that the value of /^ diminishes as m incroaaoa, for 8iii"'*'>x la laaa thaa sin"**. Hence and if we replace Isp + if hp^ hp-\ by their values from the formula abort, wa find the new inequalities 2 ' 2p + 1 where we have set, for brevity, 2 2 4 4 2p-2 «p '^ \'z'^' b"2p-\'%p-\' It is evident that the ratio ie/2Hp approachea the limit one aa p nitely. It follows that n/2 is the limit of the product H^ aa tha nombar of factors increases indefinitely. The law of formation of the snooaarita teotoca ia apparent. 117. The integral /cos (ax + b) COS (a'x + b')...djL Let as a product of any number of factors of the form oos(aa; + *X ^*»*«* a and b are constants, and where the same factor may ocour times. The formula C08(M 4- tt) . COS(li - v) cos u cos t; = ^^ ^ + ;, 242 INDEFINITE INTEGRALS [V,§117 enables us to replace the product of cwo factors of this sort by the sum of two cosines of linear functions of x ; hence also the product of n factors by the sum of two products of 71 — 1 factors each. Repeated applications of this formula finally reduce the given inte- gral to a sum of the form 1,H cos(Ax + B), each term of which is immediately integrable. If A is not zero, we have J'cOB(^:. + J)<fe= ^'"(-^^ + ^> + C, while, in the particular case when ^ = 0, /cos B dx = x cos B -{- C. This transformation applies in the special case of products of the form cos"* a; sin" a;, where m and n are both positive integers. For this product may be written cos'"aj cos" (f-)' ftnd, applying the preceding process, we are led to a sum of sines and cosines of multiples of the angle, each term of which is immediately integrable. As an example let us try to calculate the area of the curve (1)'^©' which we may suppose given in the parametric form x = a cos*^, y = 6 B\n*$, where varies from to 27r for the whole curve. The formula for the area of a closed curve, A = -| xdy — ydx, A=J -^-sin^dcos^drfd. But we have the formula (sin e cos ^« = 1 8in«2d = ^ (^ - ^^^ ^^) • Heooe the area of the given curve is A-^f^L sin4d"|"' Sirah V,§117J TRANSCENDENTAL FUNCTIONS UM It is now easy to deduce the following formulae *. I Bin X ox— I 2 *** "2 4 — ^ ^» /. , . rSsinx — 8in3ar , 3ooez . ooe3a; fcos'xdx- fl±^^^dx „ ^ I «^° 2g /- _ r 3 COS r 4- COS 3x , 3 sin X . sin 3x _ C08*xdx=J ^^ dx =__ + __ + c, /. , r3-|-4cos2x + co8 4x , 3a; . sin2x . sinix . ^ A general law may be noticed in thesr foMini];»-. I h,' ii.t.-n!- F(x) = J sh\''xdx and <I>(a') = T cos"x^/x have th«' jM-rnxi L'tt when n is odd. On the other hand, when n is even, theee integrals increase by a positive constant when x increases by 27r. It is en- dent a priori that these statements hold in general. For we hare J pin ptw + s I sin^xrfxH- I sin'xdlz, Jtw or ' sin-o-rfx-f / sin"xrfx = F(x)4- / sin'x(/T. since sin x has the period 27r. If n is even, it is evident that the integral j'^'sin^'x rfx is a positive quantity. If n is odd, the same integral vanishes, since sin (x 4- tt) = — sin x. Note. On account of the great variety of trausformations appli- cable to trigonometric functions it is often convenient to iDtrodtiM them in the calculation of other integrals. Consider, for example, the integral f 11/(1 -{■ x^)^]dx. Setting a;«tan^, this integral becomes / cos <(> d<t> = sin <f> -{- C. Henoe, returning to the variable % /; ^ ^ +c, (1 4- a-«)« VlTl? which is the result already found in % 105. S44 INDEFINITE INTEGRALS [V, §118 118. The integral /R(x)e^dx. Let us now consider an integral of the form j lt{x)e'^dx, where R{x) is a rational function of x. Let u« suppoee the function R{x) broken up, as we have done times, into a sum of the form where E{x), A^y .4,, .-, A^, J^i, •••, ^p are polynomials, and Z, is prime to its derivative. The given integral is then equal to the sum of the integral f E(x)e^''dx, which we learned to integrate in f 85 by a suite of integrations by parts, and a number of integrals of the form "A e'^'dx i- There exists a reduction formula for the case when n is greater than unity. For, since X is prime to its derivative, we can determine two polynomials X and /* which satisfy the identity ^ = XZ + ^Z'. Hence we have and an integration by parts gives the formula j^^"iF = -;r3i]^+;ririj \n-. dx. Uniting these two formulae, the integral under consideration is reduced to an integral of the same type, where the exponent n is reduced by unity. Repeated applications of this process lead to the integral Vie"' dXf X where the polynomial B may always be supposed to be prime to and of less degree than A'. The reduction formula cannot be applied to this Integral, but if the roots of X be known, it can always be reduced to a single new type of transcendental function. For definiteness suppose that all the roots are real. Then the integral In quasticm can be broken up into several integrals of the form J x^a dx, X — a V.jnyj TRANSCENDENTAL FUNCTIONS t46 Neglecting a constant factor, the substitutions s m a 4- y/«, uwm^ enable us to writ., tins integral in either of the fMlV»witjg forms: !'-?■ /iS u The latter integral / [1/log u'\du is a transoendental fuDOtion whidi is called the integral logarithm. 119. Miscellaneous integrals. Let us consider an integral of the form f' where / is an integral function of sin x and ooe x. Any tflrm of this integral is of the form I e"8in'"xco8"a;rfx, where m and n aie positive integers. We have seen above that the product sin*" a; cos^x may be replaced by a sum of sines and cotinet of multiples of x. Hence it only remains to study the foUowiny two types : I e"' cos bxdx, j ef" sin hx dx. Integrating each of these by parts, we find tne lormulsB /, , e^'sin^x o, C ^ . . , /. , , e^coste , o r^ . . e*" sin bxdx = "^ 7 / ^ '*'*'* ^ ^' Hence the values of the integrals under consideration are /, , «"(aco8*x + *sinte) c" cos bxdx = — ^ , ,, '» / . . , «"(asm&c — dcoete) ef^ sm bxdx = — ^ TTTi ' Among the integrals which may be reduced to the preceding types we may iiHMition the following j /(log x) X- Jx , / A'^^ sin x) c/x , j/{x) arc sin X <ix, J A*) •«? tan x Af, 24ti INDEFINITE INTEGRALS [V, Exs. wh«re / denotes any integral function. In the first two cases we should take log x or arc sin a: as the new variable. In the last two we should integrate by parts, taking f{x) dx as the differential of another polynomial F(x), which would lead to types of integrals already considered. EXERCISES 1. E?alaate the indefinite integrals of each of the following functions : 1 1 z* - g» - 3a;g - a; 1 + vT+x («• + !)«' X(X» + 1)«' (X2+1)8 ' \-y/i ' 1 l+vTTx 1 + X + VH- x« i-^J/TTx' V« + Vx + 1+ Va;(x + 1)' cos^x' N 2. Find the area of the loop of the folium of Descartes : x» + y8 - Zaxy = 0. 8. Eyaluate the integral /y dx, where x and y satisfy one of the following IdenUties: (af«-a«)«-ay2(2y^3a) = o, y«(a-x) = x8, y {x^ + y^) = a {y^ - x'^) . 4. Derive the formulaB r8in-ixcos(n+l)xdx= ^^nnxcosnx J n ' /.in-ix8in(n + l)x(te= "^°"^J^"^ + c, rcos-»xcos(n+l)xdx= 2^!!^f!?^^^ /co«-ix8ln(n + l)xdx = - ^^^:i£^^!i2^ + C. 5. EvaluatA each of the following pseudo-elliptic integrals : r (n-x»)dx r (i-xa)dx J (i-a!«)vr+^* J (n-x2)vrT^' e. RMlooe the following integrals to elliptic integrals : B(«)dx / Va(l + «•) + te(i + x«) + cxa(l + x2) -f dx«* /. ^(«)dx Va(l + a^) + 6«*(l + x*) + cx*' Jl(s) draotM a ratioiua funcUon. v.Exi.] exkkcisf:s 047 7*. Let a, 5, c, d be the roou oi an equauon of tbe fooith degrte t\x) ^ a Then there exist three involutory relatioiw of tha form whioh hiterchange the roots by pain. If the rational fanetloD /fz) rritltHM Iha identity ^'>^x/(-^7^;)-. the integral f[/{x)/VP{x)]dx is peeudo-ellipUc (Be« BuU$Un de la SociMt maiique, Vol. XV, p. 106). 8. The rectification of a curve of tbe type y = Azf^ leads to an imeyral of a binomial differential. Discmui the cases of integrablUty. 9. If a > 1, show that I X 1 {a-x)VT^^ Vcfi^ Hence deduce the formula X ^^ x^*dx ^ 1.8.6.»»(2it-l) 1 Vl-«* 2.4.6..2n 10. Ji AC - B^>0, show that X ■*^* dx 1.8.6. »(2n- 3) .A--* (^x2 4. 2Bx + C)" 2. 4. 6. -.(211 -2) (^C-B«)"'^l [Apply the reduction formula of § 104.] 11. Evaluate the definite Integral 8in> zdx X ^ l + 2aco«z + a* 12. Derive the following formulte ; J_i VI - 2ax + a« VI - 2^« + ^ V^ \\-Vafi/ X * (l-az)(l -^z)dx M t-afi Li (l-2a« + <r«)(l-2/J« + ^Vrn?"l l-<*^* IS*. Derive the formula X where m and n are positive integers (m<i»). [Break ap the partial fractions.] 248 INDEFINITE INTEGRALS [V, Exs 14. Flom the prooeding exercise deduce the formola X *''^Ll^ = -!L., o<a<l, 1 + X Bin a* 1ft. Setting Ip^^ s: f t' {t + lydt, deduce the following reduction formulse : (P - 1)/-P.« = <« + M« + 1)'-'' - (2 + g - j))I-p + i,<„ aod two anAlogouB formulse for reducing the exponent q. IB. Derive formulffi of redaction for the integrals ■ J V-^x« + 25x + c' "* J (X - a)*" V^x2 + 25« + C * 17*. Derive a reduction formula for the integral r' x"dx dedace a formula analogous to that of Wallis for the definite integral X ' dx la Hm the definite integral X dx 1 + «* sin^x a floite value f 19. Show that the area of a sector of an ellipse bounded by the focal axis and a radius vector through the focus is f r do, 2 Jo (l+ccosa,)5 I p danotM the parameter b^/a and c the eccentricity. Applying the gen- eral method, make the substitutions tan w/2 = <, i = m V(1 + e)/(l - e) succes- ilfelj, and show that the area in question is A = a6 ( arc tan u - e — - — J . \ 1 + uV Aleo ihow that this expression may be written in the form A = _(0--esin0), # le the ecoentrio anomaly. See p. 406. ». Find the carves for which the distance NT, or the area of the triangle mttT, M ooaetaat (Fig. 3, p. 31). Construct the two branches of the curve. f Licence, Paris, 1880; Toulouse, 1882.] V, ExMJ EXERCISES 849 21* Setting derive the recurrent formula ^ ox From tbifi deduce the forniuln where rTsp , Vtp, U-ip^i, Vtp + 1 are polynomial! with intagral o o Mfciwte , tad where I/^p and U^p + 1 contain no odd powers of z. It !■ rHMlilj thown that these formula) hold when n = 1, and the general caae followa from th« ahof* recurrent formula. The formula for A-j^, enables us to show that n* it inooDiDMiaorahl*. Dor If we assume that n'^/i - h/a, and then replace x by %/% in Atp^ w» obcain ft relation of the form Hj = aPx/^— ^^^^ r\l-*»)«''oot^d«, \a2.4.0...4p Jo a where //i is an integer. Such an equation, however, is impoMiUe, for the ilghlF naud side approaches zero as "p increases indefinitely. CHAPTER VI DOUBLE INTEGRALS L DOUBLE INTEGRALS METHODS OF EVALUATION GREEN'S THEOREM 120. Continuous functions of two variables. Let z = f(x, y) be a function of tlie two independent variables x and y which is contin- nous inside a region A of the plane which is bounded by a closed contour C, and also upon the contour itself. A number of proposi- tions analogous to those proved in § 70 for a continuous function of a single variable can be shown to hold for this function. For instance, fftven any positive number e, the region A can be divided into tuhreffums in such a way that the difference between the values of z at any two points (x, y), (x', y') in the same subregion is less than c. We shall always proceed by means of successive subdivisions as follows : Suppose the region A divided into subregions by drawing parallels to the two axes at equal dis- tances S from each other. The corre- sponding subdivisions of A are either squares of side 8 lying entirely inside C, or else portions of squares bounded in part by an arc of C. Then, if the prop- osition were untrue for the whole region A, it would also be untrue for at least one of the subdivisions, say Jj. Sub- dividing the subregion Ai in the same manner and continuing the process indefinitely, we would obtain a sequence of squares or portions of squaies A, A^, .••, a^, ••, for which the proposition would be untrue. The region A^ lies between the two lines x = a^ and x = i,, which are parallel to the y axis, aod the two lines y = c., y = d^, which are parallel to the x axis. At n increaees indefinitely a, and b^ approach a common limit X, ftDd ^ and <^ approach a common limit ^, for the numbers a„, for example, never decrease and always remain less than a fi*ed Dumber, it follows that all the points of A^ approach a limiting 260 r 1 1 1 \jrmk\ 1 --2^---"^- ' j. 7 z Flo. 23 VI,§iJi)J INTRODUCTION GHEEN'8 TUJBOREM 2BI point (X, fx) which lies within or upon the oootoar C. The i«it of the reasoning is similar to that in § 70 ; if the theorom italad w«c« untrue, the function /(x, y) could be shown to be diecontiniMMie ai the point (K, fi), which is contrary to hypotheeie. Corollary. Suppose that the parallel lines have been rhnem so near together that the difference of any two values of < in any one subregion is less than c/2, and let i; be the distance between the successive parallels. Let (x, y) and {x'y yO be two points inside or upon the contour C, the distance between which is less than ^, These two points will lie either in the same subregion or else in two different subregions which have one vertex in Aft^tT)^ in either case the absolute value of the difference /(^, y) -A'', y") cannot exceed 2€/2 = e. Hence, yiven any positive number «, anofhrr positive Jiumber rj can be found such that \/(x, y)-f(x\ y')|<c whenever the distance between the two points (x, y) and (x\ y*), wkitk lie in A or on the contour C, is less than rf. In other words, any funo* tion which is continuous in A and on its boundary C is im(/brai/y co?itinuous. From the preceding theorem it can be shown, as in f 70, that function which is continuous in A (inclusive of its boundary) is m sarily Jinite in A. If M be the upper limit and m the lower limit of the function in A, the difference Af — m is called the oteillation. The method of successive subdivisions also enables us to show that the function actually attains each of the values m and M at least once inside or upon the contour C. Let a be a point for which saw and b a point for which z — 3/, and let us join a and 6 by a broken line which lies entirely inside C. As the point (jr, y) describes this line, z is a continuous function of the distance of the point («, y) from the point a. Hence z assumes every value fi between m and M at least once upon this line ($ 70). Since a and b can be joined by an infinite number of different broken lines, it follows that the function f(x, y) assumes every value between m and if at an infinite number of points which lie inside of C A finite region A of the plane is said to be less than / in all its diihensions if a circle of radius / can be found which entirely encloses .1 . A variable region of the plane is said to be infinitesimal 25i 1X>UBLE INTEGRALS [VI, §121 in all its dimensions if a circle whose radius is arbitrarily preas- •igned can be found which eventually contains the region entirely within it For example, a square whose side approaches zero or an ellipse both of whose axes approach zero is infinitesimal in all its dimensions. On the other hand, a rectangle of which only one side approsches zero or an ellipse only one of whose axes approaches zero is not infinitesimal in all its dimensions. 121. Doable integrals. Let the region A of the plane be divided into subregions aj, a,, • • •, a^ in any manner, and let a>, be the area of the subregiou a<, and A/,- and rrii the limits of f(x, y) in a.-. Consider the two sums = ^^iM^, «=X of which has a definite value for any particular subdivision of A, None of the sums *!?-are less than wli,* where fi is the area of the region A of the plane, and where m is the lower limit of /(ic, y) in the region A ; hence these sums have a lower limit /. Likewise, none of the sums s ^xq greater than iV/12, where M is the upper limit ofJXxy y) in the region A ; hence these sums have an upper limit /'. Moreover it can be shown, as in § 71, that any of the sums »S is greater than or equal to any one of the sums s ; hence it follows that If the function /[x, y) is continuous, the sums S and s approach a oommon limit as each of the subregions approaches zero in all its dimensions. For, suppose that i; is a positive number such that the oscillation of the function is less than « in any portion of A which is less in all its dimensions than -q. If each of the subregions %, Oti * "I a. be less in all its dimensions than -q, each of the differences Jfi — iHi will be less than c, and hence the difference S — s will be lati than tO, where O denotes the total area of A. But we have S'-M^s-i + i-r + r-s, where none of the quantities S — I, I - i\ i' — s can be negative. Haooe, in particular, / — /'<cO; and since c is an arbitrary posi- tiTe number, it follows that / = /'. Moreover each of the numbers S — / and / — # oan be made less than any preassigned number by •"/t». If) ba oocutant *. If = m = IT, = m, = *, and 5 = «= mO= ifO.- TtkAM*. VI, §121] INTRODUCTION GKKKN'S TH£ORKM S6I a proper choice of c Hence the Bums S and s Lave a ftftmnion limit /, which is called the double integral of the function /(z, y) over the region A. It is denoted by the symbol y)dxd,j^ and the region A is called t\\e field of integration. If (^.» Vi) ^ ^y P^"^* inside or on the boundaiy of the ftib- region a„ it is evident that the sum 2/(^o 17,) ». lies between the two sums S and 8 or is equal to one of tliein. It therefore alto approaches the double integral as its limit whatever be the method of choice of the point (^„ i;<). The first theorem of the mean may be extended without difficulty to double integrals. Let /(ar, ^) be a function which is continuous in .4, and let </>(jr, y) be another function which is continuous and which has the same sign throughout A, For definiteness we shall suppose that <^(ar, y) is positive in A. If M and m are the limits of /(x, y) in A, it is evident thiat* Adding all these inequalities and passing to the limit, we find the formula r / fip^y y) 4>(^^ y)dxdy = fif f 4^(x, y)dxdy, where fi lies between M and m. Since the function J\Xy y) assumes the value /x at a point (^, rt) inside of the contour C, we may write this in the form (1) jj /(x, y)^{x, y)dxdy =j\i, ^)jj^^^<.'^ y)dxd9, which constitutes the law of the mean for double integrals. If <^(j-, y) = 1, for example, the integral on the right> JJdxdjff extended over the region .1 , is evidently equal to the area Q of that In this case the formula (1) becomes (2) jJ Ax, y) iix dy = O/i;^, 1,) • If /(«, y) is a consunt *, w© «hAll h»re If « m «*, aad equations. The following formula holds, bofrever, with |i ■ *. — Teaw. 254 DOUBLE INTEGRALS [VI, §122 122. Volume. To the analytic notion of a double integral corre- sponds the important geometric notion of volume. Let fix, y) be a function which is continuous inside and upon a closed contour C. We shall further suppose for definiteness that this function is posi- tifO. Let S be the portion of the surface represented by the equa- tion « =A*"» y) which is bounded by a curve T whose projection upon the xy plane is the contour C. We shall denote by E the por- tion of space bounded by the xy plane, the surface S, and the cylinder whose right section is C. The region A of the xy plane which is bounded by the contour C being subdivided in any manner, let a,- be one of the subregions bounded by a contour c,-, and w,- the area of this subregion. The cylinder whose right section is the curve c^ cuts out of the surface 5 a portion s^ bounded by a curve y.- . Let p^ and P< be the points of «,• whose distances from the xy plane are a mini- mum and a maximum, respectively. If planes be drawn through these two points parallel to the xy plane, two right cylinders are obtained which have the same base co^, and whose altitudes are the limits 3/,- and m,- of the function /(x, y) inside the contour c.., respec- tively. The volumes K,- and v\ of these cylinders are, respectively, ^iMi and (i),mf.* The sums S and s considered above therefore repre- sent, respectively, the sums Sr^ and 2*^,- of these two types of cylin- ders. We shall call the common limit of these two sums the volume of the portion E of space. It may be noted, as was done in the case of area (§ 78), that this definition agrees with the ordinary concep- tion of what is meant by volume. If the surface S lies partly beneath the xy plane, the double integral will still represent a volume if we agree to attach the sign — to the ▼olomes of portions of space below the xy plane. It appears then th^t every double integral represents an algebraic sum of volumes, just as a simple integral represents an algebraic sum of areas. The limits of integration in the case of a simple integral are replaced in the case of a double integral by the contour which encloses the field of integration. 183. Evaluation of double Integrals. The evaluation of a double integral can be reduced to the successive evaluations of two simple integrals. Let us first consider the case where the field of integration •Bj the voftinM q^ a right eylindMt we shall understand the limit approached hy tka volama of a right prism of the same heiglit, whose base is a polygon inscribed in A rifkt itetlOB of the cylinder, as each of the sides of this polygon approaches zero. r^fcto diiallloB la nol aaoaasary for the argnment, but is useful in sliowing that the of volttme in general agrees with our ordinary conceptions. — Trans.] VI, § 123] IXTRODL'CTIOV r.RFFVS T)fBOR£M 8A6 is a rectangle H bouuded by Uiu straight Udm «hc^ * * X, y = yo, y = Y, where x^<X and y^O'. SuppoM ihb reeUoflt to be subdivided by parallels to the two axes r a x^, y ■" y* (i = 1, 2, , 7t ; /: = 1, 2, • •, m). The area of Uie small reotanfto 72^. bounded by the lines x = x<_„ ac = x^, y = v» _ ,. y s y^ ij Hence the double integral is the limit of the sum (;<) •">• = %%AU, v,t)(^ - - ,)(y. - y.-,). D _tf« Vt u (7 m ft V9 A a •o^ ha Pw. 24 where (^,4., r^a) is any point inside or upon one of tlur sides of Ri^. "We shall employ the in de- termination of the points iiiki Vik) ^^ order to simplify the calculation. Let us re- mark first of all that if f(x) is a continuous function in the interval (a, b), and if the interval (a, b) be subdivided iu any manner, a value |, can be found in each subinterval (x^ ,. r^^ §urh that (4) j[/(x)^=/(«(aH-«)+/l^O(*t-xO+--.+/(«(*-«^-0. For we need merely apply the law of the mean for int«graU to each of the subintervals (a, x,), (xj, «,), • • •, («,_„ b) to find these Tmluea of t • Now the portion of the sum S which arises from the row of reo- tangles between the lines x = Xi_| and x = x^ is (Xi - a^.-i)[/(^.n ViiXyi - Vo) H-/(6f , ViiXsft - yi) + • • • Let us take ^., = ^,., = • • • = ^<. = x,.„ and then choose i^u %,, ••• in such a way that the sum ~~ /(a:<-„ i7n)(yi - yo) +/[^<-i. %f)(yi - yO + • • is equal to the integral XV(^<-i» y)<'y» where the integral i« to be evaluated under the assumption that x<., la a ooostant If we pro- ceed in the same way for each of the rows of reotanglee bounded by two consecutive parallels to the y axis, we finally find the eqoatioo (5) 5 = *(Xo)(xi-Xo) + *(a',)(x,-x.) + ... + *(«,.0(«i-«,.0 + ---» X 256 DOUBLE INTEGRALS [VI, §123 where we have set for brevity 4<x)= ff(x,y)dy, Thia function <l>(a:), defined by a definite integral, where x is con- ■idered as a parameter, is a continuous function of x. As all the intenrals «< — «(_! approach zero, the formula (5) shows that S i^proaches the definite integral ^(x)dx. Hence the double integral in question is given by the formula W rr /(^' y)dxdy= f dx ff(x, y)dy. In other words, in order to evaluate the double integral, the function JXZf y) should first he integrated between the limits y^ and F, regard- ing X as a constant and y as a variable; and then the resulting func- tion^ which is a function of x alone, should he integrated again hetween the limits x^ and X. If we proceed in the reverse order, i.e. first evaluate the portion of S which comes from a row of rectangles which lie between two cooaecutive parallels to the x axis, we find the analogous formula / I A^i y)dxdy = I dy I f(x, y)dx. A comparison of these two formulae gives the new formula ' dx I fix, y)dy=\ dy f(x, y)dx, which is called the formula for integration under the integral sign. An MMntial presupposition in the proof is that the limits Xq, X, y^, Y are ooutants, and tliat the function f(x, y) is continuous throughout the field of integration. I^t z = xy/a. Then the general formula gives VhiiTsq INTRODUCTION 0RESN*8 THEOBUC 167 In general, if the function /(x, y) is the product of a fimofeioii ol m alone by a function of y alone, we shall hare The two integrals on the right aie absolutely independent of eaeh other. Franklin * has deduced from this remark a Tery cimple demoiiKiBtioo ol e«r- tain interesting theorems of Tchebycheff. Let ^(x) and f (x) be two fa which are continuous in an intenral (a, b), where a < 6. Then the double I ffM^) - *(y)] M"^) - nv)]dxdv extended over the square boutidod by the Unee z = a,z = 6, y = a,yB6is to the difference 2(6- a)j^ 0(x)^(x)(te - 2f\(x)dx x j\{z)dz. Bat all the elements of the above double integral have the mido sigB if the two tunctions 0(x) and ^(x) always increaee or decrease aimaltaneooaly, or If om of them always increases when the other decreasee. In the flntcaae tho twofono- tions <t>{x)-4>{y) and ^(x) - ^(y) always have the Mine sign, whenas ib^ haw opposite signs in the second case. Hence we shall have (6 - a)j\{x)i^(x)dx >f\(x)dx xf%{x)dx whenever the two functions 0(x) and ^(x) both increase or both decrease throQfh- out the interval (a, b). On the other hand, we shall have (6 - a)fy{r)rl^{x)dx < £i^{x)dx x^V(«)*t whenever one of the functions increases and the interval. The sign of the double integral is also definitely detemUaad tn oan ^) m f{z)^ for then the integrand becomes a perfect sqoare. In this case w« ahaU have (6 - a)f\i.{x)ydz > [ j]Vx)dr]', whatever be the function 0(x), where the sign of equality can hold only wh«i 0(x) is a constant. The solution of an interesting problem of the oaleolas of variatioas bmj be deduced from this result. Let P and Q be two flzod points in a plaae whose coordinates are (a, A) and (6, B), respectively. Let y =/[*) ^ the eqjaatloB of any curve joining these two points, where /(z), together with Hi flitt dwlt ell fa •American Journal qf MathBmatim^ VoL VII, p. 77. 258 DOUBLE INTEGRALS [VI, § 124 /'(x), if suppoied to be continuous in the interval (a, b). The problem is to flDd that one of the curves y=f{x) for which the integral J^*y' 2^3; ig a minimain. But by the formula just found, replacing <p{x) by y" and noting that /{a) = A and /{h) = B by hypothesis, we have (6 a)fy'^dx^{B-A)^. V A / ^ B' 1 ^A B "^ k ^ ^ 1 ! p \ ! "^ • X •-1 «i : C a Fio. 25 TTie minimum value of the integral is therefore {B - A)^/{h - a), and that value It actually a»umed when y' is a constant, i.e. when the curve joining the two Used poinu reduces to the straight line PQ. 124. Let us now pass to the case where the field of integration is bounded by a contour of any form whatever. We shall first suppose that this contour is met in at most two points by any parallel to the y axis. We may then suppose that it is composed of two straight lines X — a and x = h (a<b) and two arcs of curves APE and A'QB' whose equations are Yi = <fii(x) and Y^ = <t>2(x), re- spectively, where the functions <t>i and <^2 are continuous be- tween a and b. It may happen that the points A and A' coin- cide, or that B and B' coin- cide, or both. This occurs, for instance, if the contour is a convex curve like an ellipse. Let us again subdivide the field of integration R by means of parallels to the axes. Then we shall have two classes of subregions : regular if they are rectangles which lie wholly within the contour, irregular if they are portions of rectangles bounded in part by arcs of the contour. Then it remains to find the limit of the sum 5 = 2/(^,^)0,, where « is the area of any one of the subregions and (f, 7;) is a point in that subregion. Let us first evaluate the portion of S which arises from the row of subregions between the consecutive parallels a; = a;,_i, x = «,. These subregions will consist of several regular ones, beginning with a Yertex whose ordinate is y' > Y^ and going to a vertex whose ordinate is y" < K,, and several irregular ones. Choosing a suitable point (t 17) in each rectangle, it is clear, as above, that the portion of S which comes from these regular rectangles may be written in the form VI, §124] INTRODUCTION GREEN'S THEOREM 269 Suppose that the o8(;illation of each of the functioiui ^t(*) »o<i ^iC*) in each of the intervals (ar^_), x^ is less than 3, and that each of the differences y^ — y^_, is also less than & Then it is easily seen that the total area of the irregular subregions between x « x,., and x ss & is less than 48(x, —x^_{)y and that the portion of S which artset from these regions is less than 4//£(a:| •> ob^.,) in absolute value, where // is the upper limit of the absolute value of /|[z, y) in the whole field of integration. On the other hand> we have A^i^u y)dy=^\ /(ar,_„ y)dy{' \m+ I ., and since | K, — if\ and | Y^ — y"| are each less than 2S, we may write f A^i-iy y)dy =J^ f{x,,u y)dy + iHxi. |X«1. The portion of S which arises from the row oi subregions under consideration may therefore be written in the form (^. - ^.-i) [J^ /(Xi.i, y)dy + 8W,«J, where d, lies between — 1 and -\- 1. The sum 8^SS^«(Xj — X|.|) is less than ^Hh(b — a) in absolute value, and approaches sero with ^ which may be taken as small as we please. The double integral is therefore the limit of the sum where ♦(«)= fk^>y)dy' Hence we have the formula (7) / / /(^, y;"^ '^y = f dx f f(x, y)dy. In the first integration x is to be regarded as a oonitani, bat the limits Yi and }\ are themselves funotiona of x and not constants. 260 DOUBLE INTEGRALS [VI, §124 Ji^mplo Let iw try to evaluate the double integral of the function xy/a m inlerior of a quarter circle bounded by the axes and the circumference x2 + ya_ij2 = o. Tl>e UmlU forx are and i2, and if « ifl constant, y may vary from to VJB* - x^. theintagnlis ^-jr>.Mo^--=X *x(B^-x^)^_ a " J, 2a L Jo J^ 2a The Tilue of the latter integral is easily shown to be R*/Sa. When the field of integration is bounded by a contour of any form whatever, it may be divided into several parts in such a way that the boundary of each part is met in at most two points by a parallel to the y axis. We might also divide it into parts in such a way that the boundary of each part would be met in at most two points by any line parallel to the x axis, and begin by integrating with respect to X. Let us consider, for example, a convex closed curve which lies inside the rectangle formed by the lines x = a^ x = bj y = c, y = d, upon which lie the four points AjB, C, D, respectively, for which x or y is a minimum or a maximum.* Let y^ = 4>i(oc) and 3/2 = <^2(^) be the equations of the two arcs ACB and ADB, respectively, and let X, s= ^i(y) and «, = ^tO/) he the equations of the two arcs CAD and CBDf respectively. The functions <t>i(x) and <f>2(x) are continu- OQi between a and i, and ipi (y) and \j/2 (y) are continuous between c and (L The double integral of a function /(a;, y), which is continuous inside this contour, may be evaluated in two ways. Equating the ralues found, we obtain the formula dx f(x,y)dy= / dy f(x, y)dx. It is clear that the limits are entirely different in the two integrals. Erery convex closed contour leads to a formula of this sort. For •lample, taking the triangle bounded by the lines y = 0, cc = a, y ia X as the field of integration, we obtain the following formula, which is due to Lejeune Dirichlet : jf dx^Jix, y)dy=j^dyjf(x, y)dx. *TlM reader U advised to draw tlie figure. Vi,5125j INTRODUCTION GRKEN'S TBEOEEM SSI 125. Analogies to simple iatsfrala. The iatflgnl f'/{t^M^ Tini^ilind M % function of z, has the deriTAtiTe /(z). There eziete aa ^^•^'r^ arm Ibeons te double integrals. Let /(z, y) be a funotion which to oootiaoooi taMlde a m»» tangle bounded by the straight lines x^a^x^ A,y ^b,ym B,{(^ <A^h<B^ The double integral of /(z, y) extended over a rectaofia >*i*n fMitil by tiM ItaMS x^a,x = X,v = b,y- y,(a<J:<^, 6<r<J3),toafiiii0iloBof IkaMfli^ nates X and Y of the variable comer, that is, F(X.r)=^'*dzj[7(z,y)dy. Setting 4>(z) =^ /(z, v)dy, a first differentiation with respect to X ^^m A second differentiation with respect to Y leads to the fomnila *«' ^-|^=^'•^>• The most general function u{X, F) which tatliflM tba eqnatkm (0) to tfl- dently obtained by adding to F(.Y, Y) a funotion % wboaa nooad <torivatlf« d'^z/dXiiY is zero. It is therefore of the form (10) M(X, F) = fdz r7(z, y)dy + ^(X) + ^(F), where 0(X) and ^( F) are two arbitrary functions (see 1 88). The two aitltraiy functions may be determined in such a way that «(X, F) red o cea to a fivta function r(F) when A" = a, and to another given fnaetion Cr(X) ahl Ts li Setting AT = a and then F = & in the preceding e(|Qattoo, we obtain tha two conditions V{Y) = 0(a) + f (F), Cr(X) = *(X> + ^^(6). whence we find ^(F) = V(J) - 0(a) , ^(6) = 7(6) - ^(a) . ♦(X) = ir(X) - r(k) + #(«), and the formula (10) takes the form (11) t^(X. F) = j^^dz j^7(z, y)dy + ir(X) + F(F) - 7(6) . Conversely, if, by any means whatever, a fnnetion ii<X, F) baa bai which satisfies the equation (0), it is easy to show bj methods ilnUar to lAt above that the value of the double integral to given Uy tba formula (12) /'iix/ M V)dv = u(X. F) - u{X, 6) - u{a, Y) + «(a, 6). This formula i.s analogous to the fondamental fononln (6) on The following formula is in a sense analofoos to ttot foimito for by parts. Let ^ be a finite region of the plane boondad bj ons or i 262 DOUBLE INTEGRALS [VI, §126 of any form. A function /(x, y) which is continuous in A varies between its mlnlmam «« and ita maximum V. Imagine the contour lines /(x, y) = v drawn whU9 9 Uas between »o &Qd F, and suppose that we are able to find the area of the portkOB of A for which /(x, y) lies between vq and v. This area is a func- tion /t*) ^tiioh increases with o, and the area between two neighboring contour llMtle ^(v + Ae) — F{v) = AoF'(o + ^Ao). If this area be divided into infinitesi- ■ftl poitioni by lines joining the two contour lines, a point ({, rj) may be found In eaeh of tbem such that /((, 17) = v + SAv. Hence the sum of the elements of the double integral Jf/dx dy which arise from this region is (r + 0Av) F\v + dAv) At>. It follows that the double integral is equal to the limit of the sum 2(t> + ^ At)) F\v + ^ Ao) At) , that is to say, to the simple integral J t» F\v) dv = VF{ y)-f FCo) dv . This method is especially convenient when the field of integration is bounded by two contour lines /(x, y) = »o, /(x, y)= V. For example, consider the double integral J f y/\ + x^ -\-y^dxdy extended over the interior of the circle x^ -f- y2 _ j. if ^^ set v = Vl + x'-^ + y'^, the field of integration Lb bounded by the two contour lines v = \ and v = V2, and the foncUon F{v), which is the area of the circle of radius Vt)^ - 1, is equal to «(•• - 1). Hence the given double integral has the value j^ '2>ro«d« = ^(2%^-l). * The preceding formula is readily extended to the double integral fff(x,y)<P{x,y)dxdy, F{v) now denotes the double integral //0(x, y)dxdy extended over that of the field of integraUon bounded by the contour line v = /(x, y). ^ lae. Green»e theorem. If the function /(a;, y) is the partial deriva- tive of a known function with respect to either x or y, one of the tntagrations may be performed at once, leaving only one indicated mtogration. This very simple remark leads to a very important formula which is known as Green's theorem. m wT TTl*"!!!*^*"* **' '***■ "•"***** are to be found in a memoir by Catalan (/OMTMI 4s JJawHiU, 1st series, Vol. IV, p. 233) . Vl,Jii(ij IMKODUCTION GKKEN'S THKjOHEU Let us consider Brst a double integral ffdP/e^dxdy exiemM over a region of the plane bounded by a contour C, which it met in at most two points by any line parallel to the y axis (see Fig. IS^ p. 188). Let A and B be the points of C at which x is a minimum ftiH a maximum, respectively. A parallel to the y axil between Aa and Hb meets C in two points m^ and m, whose ordinates are yx ^^^ y» respectively. Then the double integral after integration with reapeei to y may be written But the two integrals ^ P{Xy yx)dx and /V(x, y^dx are line integrals taken along the arcs ArriiR and Am^B, respectlTely ; the preceding formula may be written in the form (13) fjf^a.ay = -jja.. where the line integral is to be taken along the contour C in the direction indicated by the arrows, that is to say in the positive sense, if the axes are chosen as in the figure. In order to extaod the formula to an area bounded by any contour we should proceed as above (§ 94), dividing the given region into several parts for eaeb of which the preceding conditions are satisfied, and applying the for- mula to each of them. In a similar manner the following analogous form is easily derived : (14) //g</W,=£«iy, where the line integral is always taken in the same sense. Sob- tracting the equations (13) and (14), we find the formula (15) //^■'^'^=//(S-'4) d^dy. where the double integral is extended over the region bounded by C, This is Green's formula ; ite applications are very important Just now we shall merely point out that the substitution Q = x and p = -y gives the formula obtained above (§ W) for the area of a closed curve as a line integral. 264 DOUBLE INTEGRALS CVI,S127 IL CHANGE OF VARIABLES AREA OF A SURFACE In the evaluation of double integrals we have supposed up to the present that the field of integration was subdivided into infinitesimal reetangles by parallels to the two coordinate axes. We are now going to suppose Uie field of integration subdivided by any two systems of ourres whatever. 127. Preliminary formula. Let u and v be the coordinates of a point with respect to a set of rectangular axes in a plane, x and y the coor- dinates of another point with respect to a similarly chosen set of rectangular axes in that or in some other plane. The formulae (16) x=f(uyv), y = 4>(u,v) establish a certain correspondence between the points of the two planes. We shall suppose 1) that the f unctions /(w, v) and <^(w, v), together with their first pai-tial derivatives, are continuous for all points (u, v) of the uv plane which lie within or on the boundary of a region Ai bounded by a contour Ci ; 2) that the equations (16) transform the region A^ of the uv plane into a region A of the xy plane bounded by a contour C, and that a one-to-one correspond- ence exists between the two regions and between the two contours in such a way that one and only one point of ^ j corresponds to any point of i4 ; 3) that the functional determinant A = D(f, <f>)/D(u, v) does not change sign inside of Ci, though it may vanish at certain points oi Ai. Two cases may arise. When the point (w, v) describes the con- tour C, in the positive sense the point {x, y) describes the contour C either in the positive or else in the negative sense without ever reversing the sense of its motion. We shall say that the corre- spondence is direct or inverse, respectively, in the two cases. The area O of the region A is given by the line integral Jtc xdy (C) taken along the contour C in the positive sense. In teims of the new variables u and v defined by (16) this becomes wbare the new integral is to be taken along the contour C^ in the positive §miM^, and where the sign + or the sign - should be taken VI, J 127] CHANGE OF YAiUAULBS 206 according as the correspondence is direct or inverse. Applying Green's theorem to the new integral with x » v, o ai y, /> ^fd^/bu^ Q=/a<^/ay, wefind du wlience ^-j;-^ or, applying the law of the mean to the (louriie integral, (17) 0=:±0i where (^, 17) is a point inside the contoar Ci, and Oi is the area of the region .li in the uu plane. It is clear that the sign + or the sign — should be taken according as A itself is positiTc or negative. Hence the correspondence is direct or inverse OMording as A is positive or negative. The formula (17) moreover establishes an analogy between func- tional determinants and ordinary derivatives. For, suppose that the region A 1 approaches zero in all its dimensions, all its points approacb- ing a limiting point {iiy v). Then the region A will do the same, and the ratio of the two areas O and 0^ approaches as its limit the abso> lute value of the determinant A. Just as the ordinary derivative is the limit of the ratio of two linear infinitesimals, the functional determinant is thus seen to be the limit of the ratio of two infinites- imal areas. From tliis point of view the formula (17) is the analogoo of the law of the mean for derivatives. Bemarki. The hypotheaes which we have made oonoemlng the ( between ^ and ^ 1 are not all Independent Thua, In order that tke ence should be one-to-one, it is neceaeary that A ■hookl not ehaafi ilga in Iha region ^1 of the uv plane. For, suppose that A vanishes along a oorva -n which divides the portion of A\ where A is positive from the portion where A is negative. Let us consider a small aro mi nx of 7i and a small portion of A\ which contains the arc mini- Thla portion is composed of two regions a\ and a'x which are separated by mini (Fig. 20). When the point (u, t) deacribea the fto. tS region a\, where A is positive, the point (x, y) describes a region a bounded by a coDtoor MapM, and the two mi ui pi mi and mnpin are described aimultaneooaly In the poritlve ansa the point (m, v) describes the region «(, where A Is negative, the polat (s, f) 266 DOUBLE INTEGRALS [VI, §128 ilMTiillim a regUm a' whose contour nmqr is described in the negative sense as Hi Ml 9i Hi it deaeribed in the poeitive sense. Tlie region a' must therefore eovw a part of the region a. Hence to any point (x, y) in the common part arm oorrespond two points in the uv plane which lie on either side of the line mi»i. As an ezAinpIe consider the transformation X = x,Y = y^, for which A = 2y. If the point (z, y) describes a closed region which encloses a segment ab of the X axis, it is eTident that the point (X, Y) describes two regions both of which lie above the X axis and both of which are bounded by the same segment AB of that axis. A sheet of paper folded together along a straight line drawn upon it fiTSs a clear idea of the nature of the region described by the point {X, Y). The condition that A should preserve the same sign throughout ^i is not suf- ficient for one-to-one correspondence. In the example X = x'^ — y^, Y = 2 xy, the Jacobian A = 4 («* + y*) is always positive. But if (r, d) and {R, w) are the polar coordinates of the points (x, y) and (X, Y), respectively, the formulae of transformation may be written in the form R = r^^ u = 2d. As r varies from a to 6 (a < 6) and $ varies from Oto?r + a(0<a< 7t/2), the point {R, u) describes a circular ring bounded by two circles of radii a^ and 62. But to every value of the angle u between and 2a correspond two values of ff, one of which lies between and a, the other between 7t and it -\- a. The region described by the point (X, Y) may be realized by forming a circular ring of paper which partially orerlape itself. 188. Transformation of double integrals. First method. Retaining the hypotheses made above concerning the regions A and Ai and the formulae (16), let us consider a function F(x, y) which is continuous in the region A. To any subdivision of the region A i into subregions On a^y •y a^ corresponds a subdivision of the region A into sub- regions a,, a,, • . •, a,. Let w, and o-, be the areas of the two corre- sponding Bubregrions a< and a,., respectively. Then, by formula (17), Wf = 0-. where u^ and v, are the coordinates of some point in the region a,. To this point (m„ t;<) corresponds a point x, =f(Ui, v,), y, = <^(w., v,) of the region a<. Hence, setting <I>(w, v) = F[f(u, v), <f>(u, v)], we may write <- 1 i -I whanee, pasting to the limit, we obtain the formula (.8, fin-, y)^^y -/X/C/C. "). *(«. V)] I 'j^^Uu,.. VI. 5 128] CHANGE OF \ AKlAbLlv.-> 267 Hence to perfomn a transformation in a double imUgral x amd y rh mU be replaced by their values as functions of ths new variaAlee u mmd 9 and dxdy should be replaced by \^\dude. We bare moo tlraadr how the new field of integration is determined. In order to find the limits between which the integrmttons ihould be performed in the calculation of the new doaUe integral, it ii ia general unnecessary to construct the contour C, of the new field of integration .l^ For, let us consider i« and tr m a fjaten of curvilinear coordinates, and let one of the yariaUet u and v in tha formulae (16) be kept constant while the other varies. We obtain in this way two systems of curves u = const, and v » eonst By the hypotheses made above, one and only one curre of each of iheaa families passes through any given point of the region A. Let us suppose for definite- ness that a curve of the family v = const, meets the contour C in at most two points Ml and 3/, which cor- respond to values u^ and w, of u (wj < ifj), and that each (p of the (v) curves which meets the contour C lies between the two curves v = a and V = 6 (a<b). In this case we should integrate first wm,ti with regard to u, keeping v constant and letting u rary from ti| to tzj, where xi^ and u^ are in general functions of r, and then inte- grate this result between the limits a and b. The double integral is therefore equal to the ezpreation M F[/(«, r), ^(«,r)]|Alifii, A change of variables amounts essentially to a snbdiTiaion of the field of integration by means of the two SjTStems of curres (m) and (v). Let 0) be the area of the curvilinear quadrilateral bounded by the curves («), {u -h du), (v), {v 4- dv), where du and rfr are poaitiTe. To this quadrilateral corresponds in the uv plane a rectangle whose sides are du and dv. Then, by formula (17), • = | A(^, iy)| dm «/r, wbove ^ lies between u and u + du, and iy between v and v + dv. The exprea- sion I A(u, r) | du dv is called the eUmml ^f arem in the tjwUm id 268 DOUBLE INTEGRALS [VI, §129 ooftidinates (u, v). The exact value of a> is w = J | A(w, v)\-\-€ldu dv, where c approaches zero with du and dv. This infinitesimal may be na^acted in finding the limit of the sum ^F{x, y) <o, for since A(w, v) U oontinuous, we may suppose the two (w) curves and the two (v) oarret taken so close together that each of the e's is less in ab- Boliite Talue than any preassigned positive number. Hence the abso- lute value of the sum SZ-X^, y)idudv itself may be made less than any preassigned positive number. 189. Examples. 1) Polar coordinates. Let us pass from rectangu- lar to polar coordinates by means of the transformation x = p cos <o, y = /> sin <u. We obtain all the points of the xy plane as p varies from zero to -I- oo and o> from zero to 27r. Here A = /» ; hence the element of area is p dm dp, which is also evident geometrically. Let us try first to evaluate a double integral extended over a portion of the plane bounded by an arc AB which intersects a radius vector in at most one point, and by the two straight lines OA and OB which make angles wi and wj with the x axis (Fig. 17, p. 189). Let R = ^(«) be the equation of the arc AB. In the field of integration m varies from o>i to (i>s and p from zero to R. Hence the double inte- gral of a function /(x, y) has the value \ dm\ f{ p COS o), p sin 0)) p dp. If the arc AB ia a. closed curve enclosing the origin, we should take the limits <i»i = and 0)3 = 27r. Any field of integration can be divided into portions of the preceding types. Suppose, for instance, that the origin lies outside of the contour C of a given convex closed curve. Let OA and OB be the two tangents from the origin to this curve, and let R^ =fi(ui) and R^ =/2(<o) be the equations of the two arcs ANB and A MB, respectively. For a given value of w between <i>i and w^, p varies from i2, to R^, and the value of the double integral is f(p cos o), p sin <a)pdp. fl) EU^k eoiirdinaUM. Let ui consider a family of confocal conies VI, $ 130] CHANGE OF VARIABLKS SM where X denolM an arbitrary parameter. Tbroogh urerj point of the two conies of thia family, — an aiUpie and an hyperbola, — for Iho r w »S m OV-^^-^F has one root X greater than (^, and another posiUte root |i values of x and y. From HO^ and from the analogont replaced by m we find than ^, for any X la (20) VXm V^(X-c«)(c*-m) 0<^^<^^X. To avoid ambiguity, we shall consider only the first quadrant in tlie This region corresponds point for point in a one-to-one mannur to tba the X/i plane which is bounded by the straight lines X = c», M = 0. /» = <^. It is evident from the formulie (20) that when the point (X, f) deaBribei tko boundary of this region in the direction indicated by the arrowa, the point (x, y) describes the two axes Ox. and Oy in the aenie i ndicated by the arrowa. Tbe transformation is therefore inverse, which is verified by calcniating A : ^^ J>(x, y) ^ 1 X-n D(X, n) 4 VXM(X-r«)(c«-^)* 130. Transformation of double integrala. Second method. We ihall now derive the general formula (18) by another method which depends solely upon the rule for calculating a double integral. We shall retain, however, the hypotheses made aboTe ooooemiDg the correspondence between the points of the two regions A and Ax* If the formula is correct for two particular traasformatioos it is evident that it is also correct for the transformation obtained by carrying out the two transformations in mooession This fottows at once from the fundamental property of foBetknal detenninants (§80) D{T. y) i>(ar, y) D(u, v) D{u\ u') D{u, v) D(u\ rO S70 DOUBLE INTEGRALS [VI. § 130 Similarly, if the formula holds for several regions A,B,C, ",L, to which correspond the regions Ay, Bi, Ci, ••-, Li, it also holds for the region A -^ B -k- C -i \- L. Finally, the formula holds if the trmnsfonnation is a change of axes : X = ar^ -f x' COS a - y' sin a, y = yo + x' sin a -\- y' cos a. Here A = 1, and the equation F(Xf y) dx dy If' ■II' F(x^ 4- x' cos or — y' sin a, y^ + x' sin a + y' cos a) dx' dy' is ffttisfied, since the two integrals represent the same volume. We shall proceed to prove the formula for the particular trans- formation (21) x = <^(x', y'), y = y which carries the region A into a region ^' which is included between the same parallels to the x axis, y — yo and y =zy^. We shall sup- pose that just one point of A corresponds to any given point of A ' and conversely. If a paral- lel to the X axis meets the boundary C of the region A in at most two points, the same will be true for the boundary C" of the region yl'. To any pair of points itiq and mi on C whose or- dinates are each y cor- respond two points w'o mjr frriA /ffi] s: Tnp; Fio. 29 ■ad ml of the contour C. But the correspondence may be direct or tnTsrse. To distinguish the two cases, let us remark that if d<l>/dx' is positire, x increases with x', and the points mo and m^ and ?wi and *h' lie as shown in Fig. 29 ; hence the correspondence is direct. On the other hand, if dif>/dx' is negative, the correspondence is inverse. Ut us consider the first case, and let Xo, Xj, x^, x| be the abscissse of the points m,, m,, w;, mj, respectively. Then, applying the for- mula for change of variable in a simple integral, we find jf V y)dx =jr F[^(x', y% y'] grfx', VI, §130] CHANGE OF VABUBLE8 271 where y and y' are treated as oomtonta. A lingle intA^frrAiion girtt the formula ^ dyj\x, y)dx =f\'f'nH'\ y'), yl Ijrf''. But the Jacobian A reduces in this case to d^/9x\ aod hence Uie preceding formula may be written in the form ff F(x, y)dxdy = ff F[^(x', y*), y'] } A | rfx'rfy'. This formula can be established in the same '"^nnfw if d^fdx' is negative, and evidently holds for a region of any form whatever. In an exactly similar manner it can be shown that the trans- formation (22) x = x\ y = ^(x-, yO leads to the formula / / ^\^y y)dxdy=\\ F[x', ^(x\ y')]|Ai<ix'c/y', J J(A) J J(A') where the new field of integration A* oonesponds point for point U* the region A. Let us now consider the general formulae of transformatioo (23) «=A«i, yi), y=/i(afi, yi), where for the sake of simplicity (x, y) and (Z], yi) denote the coor- dinates of two corresponding points m and My with retp»c t to the same system of axes. Let A and .-1 1 be the two oorresponding regioot bounded by contours C and Ci, respectively. Then a third point m\ whose coordinates are given in terms of those of m and Af, by the relations x' = x^j y' = y, will describe an auxiliary region A\ which for the moment we shall assume corresponds point for point to each of the two regions A and A^. The six quantities x, y, x,, yj, x*, y satisfy the four equations x==f(xi>yi)y y=/i(«i,yi), «'-*i, y-yi whence we obtain the relations (24) a:' = x„ y' =»/i(«„ y,). which define a transformation of the type (22). From the oqu:ition y' =fi(x', y,) we find a relation of tho form y, = Tr(x', y') ; hrnce we may write (26) x=/(x',yO=^(ar',y'), 272 DOUBLE INTEGRALS [VI, § 131 The given transformation (23) amounts to a combination of the two tpansformationa (24) and (26), for each of which the general formula holds. Therefore the same formula holds for the transformation (23) . Remark. We assumed above that the region described by the point m' corresponds point for point to each of the regions A and A I. At least, this can always be brought about. For, let us con- sider the curves of the region Ai which correspond to the straight lines parallel to the x axis in ^. If these curves meet a parallel to the y axis in just one point, it is evident that just one point m' of A' will correspond to any given point m of A. Hence we need merely divide the region Ai into parts so small that this condition is satisfied in each of them. If these curves were parallels to the y axis, we should begin by making a change of axes. 181. Area of a curved surface. Let 5 be a region of a curved sur- face free from singular points and bounded by a contour r. Let S be subdivided in any way whatever, let 5»- be one of the subregions bounded by a contour y,-, and let m^ be a point of s^. Draw the tan- gent plane to the surface S at the point w,-, and suppose s^ taken so small that it is met in at most one point by any perpendicular to this plane. The contour y< projects into a curve y- upon this plane ; we shall denote the area of the region of the tangent plane bounded by yl by o-^. As the number of subdivisions is increased indefinitely in such a way that each of them is infinitesimal in all its dimensions, the sum 2<r,- approaches a limit, and this limit is called the area of the region S of the given surface. Let the rectangular coordinates a;, y, « of a point of S be given iii terms of two variable parameters u and v by means of the equations (26) x=/(m, v), y = <t>(u,v), z = xf;(u,v), iu such a way that the region S of the surface corresponds point for point to a region R of the uv plane bounded by a closed contour C. We shall assume that the functions /, <^, and ^, together with their first partial derivatives, are continuous in this region. Let R be subdivided, let r, be one of the subdivisions bounded by a contour c., and let «| be the area of r<. To r< corresponds on ^ a subdivision s< bounded by a contour y<. Let or< be the corresponding area upon the \ tangent plane defined as above, and let us try to find an expression for the ratio a,/w|. I^t rto A» yt be the direction cosines of the normal to the surface S At A point m,(ar<, y<, «^) of «. which corresponds to a point (w., v,) VI, § 131] CHANGE OF VARIABLES m of r^. Let us take the point m^ aa a new origin, md m the new the normal at m, and two perpendicular lines m^X and m^K in lh« tangent plane whose direction cosines with reepect to the old asee ar« a', p\ y' and a", /3", y", respectively. Let X,Y,Zhb the of a point on the surface S with respect to the by the well-known formulae for transformation of infttrttimfctw^ we shall have J^ = a'(x-x,) + /9'(y-y,) + y'(s_i^J, Y = a'\x - X,) + ^"(y - y,) + /'(, - ,J, Z = a, (X - x,) -f A (y - y,) + y, (s - «,) . The area o-< is the area of that portion of the XY plane which is bounded by the closed curve which the point (X^ Y) deecribet, at the point (w, v) describes the contour c,. Hence, by f 127, D{X, Y) D{uU v^ where u\ and v[ are the coordinates of some point inside of r easy calculation now leads us to the form + (^'''"-''V')|^-H(«'r-/»'«.")^. <r,- — «»^ or, by the well-known relations between the nine direotioo /)(«;, rO \ "' D(ul, vl) ^ f^ D(uU r{) * ^' ZK-J, O S Applying the general formula (17), we therefon obtain the eqoatioo ^ D(ji^ Diz^l D^Ua.1 <r.= where wj and vj are the coordinates of a point of the region r^ in the uv plane. If this region is very small, the point (iij, rj) is reiy the point (m^, v<), and we may write />(s,x) ^ D(«,x) Scr^ = 2<i»i «. where the absolute value of $ does not ezoeed unilj. Bines the derivatives of the functions /, ^, and f are oontinuoos in the 274 DOUBLE INTEGRALS [VI, §131 ragion R, we may assume that the regions r.. have been taken so small that each of the quantities c<, c^, c]' is less than an arbitrarily pieassigned number 17. Then the supplementary term will certainly be lees in absolute value than SrfQ, where ft is the area of the region R. Hence that term approaches zero as the regions s, (and r<) all approach zero in the manner described above, and the sum S<r< approaches the double integral dudv, where a, ft y are the direction cosines of the normal to the surface S at the point (k, v). Let us calculate these direction cosines. The equation of the tangent plane (§ 39) is ^^ '^ D(u, v)^^^ y^ D{u, v) ^ ^ ^ D(u, v) "' whence a P y ±1 D(u, v) D{u,v) D{u, v) \Li>(w, v)J Choosing the positive sign in the last ratio, we obtain the formula ^ ^(y> ^) , q DJ^^ X) D(x, y) D{uy v) ^ D(u, v) ^ D{u, v) \Lz>(w, v)J ^ [_D(u, v)J ^ LD{u, v)J Tht well-known identity (ah' - bay + {be' - cb')* -f (ca' - acy = (a« 4- *• + c'Xa'^ + ^'' + c'^) - (aa' + bb' + cc')«, which was employed by Lagrange, enables us to write the quantity the radical in the form EG — F\ where <^ --s(^ij> F^stt' <^-s(iJ' the symbol S indicating that a; is to be replaced by y and z succes- sifelj and the three resulting terms added. It follows that the area of the faifaee S is given by the double integral (38) A =Jj y/EG - F* du dv . VI, $ 132] CHANGE OF VARIAaLBS 876 The functions E, F, and G play an important part in tha Umocj of surfaces. Squaring Uie expressions for dx^ dy, and dm and adding the results, we find (29) d8* = dx*'hdy* + dx*=E du* -f 2Fduuv -r >. dv». It is clear that these quantities /?, F, and G do not depend upon the choice of axes, but solely upon the surface S itaelf and the inde- pendent variables u and v. If the variablea ii and v and the tor- face S are all real, it is evident that EG — F* must be poiiitiv<». 132. Surface element, i ne expression vEG — F*dudv ia caliea tA$ element of urea of the surface .S* in the system of co5rdinates (m, v). The precise value of the area of a small portion of the surfa ce bounded by the curves (w), (« -|- rfu), (v), (v -f dv) is {^EG — F* + t)dud9, where e approaches zero with du and dv. It is evident, as abore, that the term c du dv is negligible. Certain considerations of differential geometry oontirm in is reeuit For, if the portion of the surface in question be thought of aa a amall curvilinear parallelogram on the tangent plane to ^ at the point («, «), its area will be equal, approximately, to the product of the lengthi of its sides times the sine of the angle between the two cunrea («) and (v). If we further replace the increment of arc by the differ- ential dSf the lengths of the sides, by formula (29), are -y/Bdm and ^/(Jdv, if du and dv are taken positive. The direction parametm of the tangents to the two curves (m) and (r) are ftc/^ ^/^ H/du and cx/dv, dy/dv, dzj^v, respectively. Hence the angle a them is given by the formula cos a •=■ A^^m whence sin a — ^EG — F*/VeG. Forming the prodnet mentioned, we find the same expression as that given above for the element of area. The formula for cos a shows that F = when and only when the two families of curves (u) and (r) are orthogonal to each other. When the surface S reduces to a plane, the formula just found reduce to the formuUe found in $ 128. For, if we set f{u, r) ■■ 0, we find -(I-:)"-©" -gif^UK- "-m'^Ci)' 276 DOUBLE INTEGRALS [VI, § 132 whence, by the rule for squaring a determinant, • dx dx du dv 2 E F dj d_i du dv F G ^^EG- F^ Hence ^EG — F* reduces to |A|w Exampleg. 1) To find the area of a region of a surface whose equa- ^^ Ig g =/(x, y) which projects on the xy plane into a region R in which the function f(x, y), together with its derivatives p = df/dx and q = df/dyy is continuous. Taking x and y as the independent vari- ables, we find F = 1 -f jo^ F=pq, G=l-{- q% and the area in ques- tion is given by the double integral (30) A =/jryrTyT7<^<^. =/X,S^' where y is the acute angle between the « axis and the normal to the surface. 2) To calculate the area of the region of a surface of revolution between two plane sections perpendicular to the axis of revolution. Let the axis of revolution be taken as the z axis, and let z =f(x) be the equation of the generating curve in the xz plane. Then the oodrdinates of a point on the surface are given by the equations x = pcos<i), 2/ = /osin(i), z=f(p)j where the independent variables p and w are the polar coordinates of the projection of the point on the xy plane. In this case we have ds^ = dp^ll + f>\p)}-{-p^d<o^, E^l + f'\p), F=0, G = p\ To find the area of the portion of the surface bounded by two plane •ections perpendicular to the axis of revolution whose radii are pi and Pf, respectively, p should be allowed to vary from pi to pa (pi< P2) and M from tero to 27r. Hence the required area is given by the integral A = jT dpj^ p v/l + /'\p)rf<u = 27rJ'^p^l+f'\p)dp, tod can therefore be evaluated by a single quadrature. If s denote the are of the generating curve, we have VI, J 133] IMPROPER INTEGRAtfi 277 and the preceding formula may be written in the form J I 2'rrp(U, The geometrical interpretation of this result it etty: 2wpd* it the lateral area of a frustum of a cone whoee lUni height U d$ and whose mean radius is p. Replacing the area between two tiH?tfffPt whose distance from each other is infiniteaimal by the Utenl aiva of such a frustum of a cone, we should obtain preciaelj the above formula for A. For example, on the paraboloid of revolution generated by revolv- ing the parabola x^ = 2pz about the z axis the area of the teotum between the vertex and the circular plane section whoee radius is r is m. GENERALIZATIONS OF DOUBLE INTEGRALS IMPROPER INTEGRALS SURFACE INTEGRALS 133. Improper integrals. Let f(x, y) be a function which is tinuous in the whole region of the plane which lies outside a contour T. The double integral of /(x, y) extended over the regioo between F and another closed curve C outside of F has a finite valuo. If this integral approaches one and the same limit no matter how C varies, provided merely that the distance from the origin to the nearest point of C becomes infinite, this limit is defined to be the value of the double integral extended over the whole regioo outside F. Let us assume for the moment that the funotion /(*, y) has a constant sign, say positive, outside F. In this case the limit of the double integral is independent of the form of the curves C For, let Cj, C,, • • •, C., • • • be a sequence of closed curves each of whieb encloses the preceding in such a way that the distance to the nearesi point of C\ becomes infinite with n. If the double integral /, extended over the region between F and C, approaches a limit /, the same will be true for any other sequence of curves C{, Ci, •••, C^, ••• which satisfy the same conditions. For, if /I, be the value of the double integral extended over the region between P and C^, n may be chosen so large that the curve C, entirely enoloees C^, and w% shall have l'<K< /. Moreover /I increases with ak Uenee il 278 DOUBLE INTEGRALS [VI, §133 hat a limit /* < /. It follows in the same manner that I < V. Hence /* =s /, i.e. the two limits are equal. Ab an example let us consider a function /(x, y), which outside a eiiiJe of radius r about the origin as center is of the form where the value of the numerator ^(a;, y) remains between two posi- tive numbers m and M. Choosing for the curves C the circles ooDoentrio to the above, the value of the double integral extended over the circular ring between the two circles of radii r and R is given by the definite integral i""X' ^(p cos (u, p sin Q>)p dp p" It therefore lies between the values of the two expressions 2-J"^. 2.Mf dp p2«-l" By $ 90, the simple integral involved approaches a limit as R increases indefinitely, provided that 2a — 1 > 1 or a>l. But it becomes infinite with R if a<l. If no closed curve can be found outside which the function f(x, y) has a constant sign, it can be shown, as in § 89, that the integral !ff{Xy y)dxdy approaches a limit if the integral Jf\f(x, y)\dxdy itself approaches a limit. But if the latter integral becomes infinite, the former integral is indeterminate. The following example, due to Cayley, is interesting. Let f{x, y) = sin (x^ + y^), and let us inte- grate this function first over a square of side a formed by the axes and the two lines x = ayy = a. The value of this integral is M^' %\T\{7*'\-y^dy %mx*dx X I cosy^dy+j C08x^dx x I sin y^dy. Am a inereases indefinitely, each of the integrals on the right has a limit, by f 91. This limit can be shown to be V7r/2 in each case ; hence the limit of the whole right-hand side is tt. On the other band, the double integral of the same function extended over the quarter circle bounded by the axes and the circle x^ 4- y^ = R* is equal to the expression VI, flWj IMPROPKK IVTFriRUs ff^ which, as R becomes infinite, osciUatat between lero tnd v/2 and does not approach any limit whatever. We should define in a similar manner the double infeegral of a function /(;r, y) which becomes infinite at a point or all along a line. First, we should remove the point (or the line) from the field of integration by surrounding it by a small contour (or by a eooUwr very close to the line) which we should let diminish indefinitely. For example, if the function J{x, y) can be written in Um fom /(x, y) = ^^>y) in the neighborhood of the point (a, b), where f (ar, y) liea twtwseu two positive numbers m and .U, the double integral of /(», y) extended over a region about the point (a, h) which contains no other point of discontinuity has a finite value if and only if a is less than unity. 184. The function B(p, q). We have .i -;: .. 1 i' . • v C, recedes indefinitely in every direction. H t r . . :. . , jp. pose that only a certain portion recedes to infinity. Thiji U tht cait example of Cayley 's and also in the following example. Lei as take /(x, y) = 4aE«J'-»y««-»«— '-»•, where p and q are each positive. This funcUon is continoous sad podtiTe in tke first quadrant. Integrating first over the square of side • boonded 1^ Um aas and the lines x = a and y = a, we find, for the value ol the doable laiefial, r"2x«i»->e-«'dx X r*2y«t-»«-i'4r. Each of these integrals approaohes a limit as a becomes inflnlle. For. by ihm definition of the function T{p) in $ 92, r(p)= fV-Je-'dC, whence, setting i = x*, we find (31) r(p)= r**ax«i-»e-'*di. Hence the double integral approaches the limit r(p)r(9) as a beooflM Let us now integrate over the quarter circle boonded by tiM ans aad I circle x^ + i^ = I{*. The value of the double iategral In polar eofltdiaalea Is 2gO DOUBLE INTEGRALS [VI, §135 Ai R becomei Infinite this product approaches the limit T(p-\-q)B(Py q), wbtn w« hATe aet w ^^ B(p, q)=f*2 cosap-i an^q-i d<t> . liyiiMlm the fact that these two limits must be the same, we find the equation (88) * T{p) nq) = T{p + q) B(p, q). The Intagral B(p, q) U called Euler's integral of the first kind. Setting t = sin^ 0, it may be written in the form (84) B(p,q)=f\^-Hl-ty-'dt. The formula (83) reduces the calculation of the function B{p, q) to the calcu- laUon of the function r. For example, setting p = q = 1/2, we find whoDoe r(l/2) = Vjt. Hence the formula (31) gives 2 In general, setting q = l-p and taking p between and 1, we find X^/i-ty-^dt \~r) J We ■hall Me later that the value of this integral is TC/smpje. Itf. tarface Integrals. The definition of surface integrals is analogous to that of line integrala. Let S be a region of a surface bounded by one or more curves r. We ahall aesome that the surface has two distinct sides in such a way that if one side be painted red and the other blue, for instance, it will be impossible to pass from the red tide to the blue side along a continuous path which lies on the sur- fiioe and which does not cross one of the bounding curves.* Let us think of S as a nftteiial torface having a certain thickness, and let m and m' be two points QMur eaeh other on opposite sides of the surface. At m let us draw that half of the DomuU mn to the surface which does not pierce the surface. The direction thof dtlfaMd i4>on the normal will be said, for brevity, to correspond to that side of Um Mr&oe on which m lies. The direction of the normal which corresponds to the other tide of the surface at the point m' will be opposite to the direction joitdtflned. Lst s s ^(x, y) be the equation of the given surface, and let <S be a region of this furfioo booaded by a contour r. We shall assume that the surface is met fft at SMMl one point by any parallel to the z axis, and that the function 0(x, y) * It li Tety eaay to form a sarfaoe which does not satisfy this condition. We need miif doform a rectangular sheet of paper ABCD by pastiuK the side BC to the side AD la tMh A way that the polot C ooincidee with A and the point B with 1). yi,im] 8UBFACE IKTE6RALS og| is continuouB inside the region il of the xy pUoe wnvQA 14 bPttfided bj ihM eime C into which r projecU. It is eTldent that this MTteot hM two ^ ts for mhkh the corresponding directions of the normal make, iwpMtifely, aesta and angles with the positive direction of the z axis. We ahall call that aid* corresponding normal makes an acate angle witli the poaHlva a ■**■ the aide. Now let P(x, ]/, z) be a function oA the three rariablea s, y, and i whkk is continuous in a certain region of space which oontaina the refkMi B of Ihaavw face. If z be replaced in this function by ^(z, y), there reaalu a eertalo foaeCkiB P [z, y, ^(z, v)]oix and y alone ; and it is natural by analogy with Um intagnito to call the double integral of this function extended orer the regioa A, (^&) ff P[x.y,^(*,y)]dxdy. the surface integral of the function P{z, y, t) taken oTer the regiioo 5 of tha glrca surface. Suppose the coordinates z, y , and z of a point of 8 giren in term ctf two auxiliary variables u and v in such a way that the portion 5 of the aufaee con^ sponds point for point in a one-to-one manner to a region A of the «t plane. Lei d<r be the surface element of the surface 8, and 7 the acute angle baHwoaa Um poii- tive z axis and the normal to the upper side of S, Then the p TrH ' n g doohla integral, by §§ 131-132, is equal to the double integral (36) C C Fix, y, z) cos -r«l<r where z, y, and z are to be expressed in terms 01 <« auu v. Thia nowi is, however, more general than the former, for coa7 may take oa either o< two values according to which side of the surface ia choaea. When the aoiifta aa|^ 7 is chosen, as above, the double integral (35) or (86) ia called the aorfaoe (37) ffp{x,v,z)dxdy extended over the upper side of the surface S. liui if > he taken as the obt4iae angle, every element of the double integral will be changed in aign, and the Dew double integral would be called the surface integral // Pctsdy ezianded orar tka lower sideofS. Ingeneral, the surface integral// Pdzdyiaeqtialfo ± tliedosbia integral (85) according as it is extended over the upper or the lowar aide o( 8. This definition enables us to complete the analogy between ainple aftd doaUa integrals. Thus a simple integral changea sign when the Umlu are intetchaafed, while nothing similar has been developed for double laiegrala. With the gen- eralized definition of double integrals, we may say that the lntafral///(z, y)d« ^ previously considered is the surface integral extended over the upper aide of the xy plane, while the same integral with iu sign changed repnaenU the antfaoe integral taken over the under side. The two senaea of motion for a dai pl e iBl»> gral thus correspond to the two sides of the xy pUna for a dooblt IntagiaL The expression (3d) for a surface integral evidently doea noi require that the surface should be met in at moat one point by any parallal to tiM f axiiL In the same manner we might define the surface integrate //g(x, y. «)dycfz, ff^^i^ y» «)dtda, 282 DOUBLE INTEGRALS [VI, § i3e and tht more gener&l integral ffP(x, y, z)dxdy-hQ{x, y, z)dydz + R(x, y, z)dzdx. Mm hitngnl may also be written in tlie form r r [P C08 7 + Q cos a + -B COB /3] d<r , a, ^, Y aie the direction angles of the direction of the normal which cor to the side of the surface selected. Sorfaoe integrals are especially important in Mathematical Physics. Its. StokM' theorem. Let £ be a skew curve along which the functions P(x, y, «), Q{x, y, z), iJ(x, y, z) are continuous. Then the definition of the line integral Pdx-\-Qdy-\-Rdz L taken along the line L is similar to that given in § 93 for a line integral taken along a plane curve, and we shall not go into the matter in detail. If the curve L Is closed, the integral evidently may be broken up into the sum of three line inte- grals taken over closed plane curves. Applying Green's theorem to each of these, it is evident that we may replace the line integral by the sum of three double integrals. The introduction of surface integrals enables us to state this result in Terj compact form. Let OB consider a two-sided piece S of a surface which we shall suppose for deftniteneas to be bounded by a single curve r. To each side of the surface oorreiponds a definite sense of direct motion along the contour F. We shall BiMime the following convention : At any point M of the contour let us draw that half of the normal Mn which corresponds to the side of the surface under concideration, and let us imagine an observer with his head at n and his feet at M) we shall say that that is the positive sense of motion which the observer must take in order to have the region S at his left hand. Thus to the two sides of the surface corre- spond two opposite senses of motion along the contour r. Let us first consider a region S of a sur- face which is met in at most one point by any parallel to the z axis, and let us suppose the trihedron Oxyz placed as in Fig. 30, where the plane of the paper is the yz plane and the x axis extends toward the observer. To the boundary r of S will correspond a closed contour C in the xy plane ; and these two curves are described simultaneously in the sense indicated by the arrows. Let the eqaaUon of the given surface, and let P(x, y, z) be a function which ii conUouout In a region of space which conUins 8. Then the line inte- Fio.ao « -P«, y) be which li oontii i^* ^n'*^'' V,t)dx\» Identical with the line integral VI, §ia«] SURFACE INTEGRALS uken along the plane curve C. Lei at applj Ora«B*a thMtwi (| 1S8) to latter integral. Setting ?(5ry) = P[«, y, ^x,|f)] for definiteneas, we find dy dy Hby"^ 9f 0017* where a, /3, 7 are the direction angles of the normal to the l^ptr iMt of 0. Heuce, by Greeirs theorem, where the double integral ia to be taken over the ngioo A of the «y |4aao bounded by the contour C. But the right-hand aide ia simply the surtaeo integral //(f-«-g»') extended over the upper side of 8 ; and henoe we may write f P(x,y,z)dx= r f t^dzdz^^dxdv. This formula evidently holds also when the surface integral is taken other side of S, if the line integral Is taken in the other directioti aloof P. Aad it also holds, a^ does Green> theorem, no matter what the foiw of tke may be. By cyclic permutation of x, y, and z we obtain the fol formulee : Adding the three, we obuin Stokm'^ lAaorcm to iU gm§nifonm : f P{x, y, x)dz + Q(x, y. t)dy + «(«, y, i)df (88) The sense in which r is described and the side of tbe eafteoe over wlOeii double integral is taken correspond aooording to tbe eoateslkHi 2g4 DOUBLE INTEGRALS rvi,§i37 IV. ANALYTICAL AND GEOMETRICAL APPLICATIONS 137. yolmnM. Let us consider, as above, a region of space bounded by the xy plane, a surface 5 above that plane, and a cylinder whose generators are parallel to the z axis. We shall suppose that the section of the cylinder by the plane « = is a contour similar to that drawn in Fig. 25, composed of two parallels to the y axis and two oorvilinear arcs A PB and A 'QB'. li z = f{x, y) is the equation of the surface S, the volume in question is given, by § 124, by the integral A^y y)dy. Now the integral C*f(^y y)^y represents the area A of a section of this volume by a plane parallel to the yz plane. Hence the preceding formula may be written in the form (39) F=jr b Adx, The volume of a solid bounded in any way whatever is equal to the algebraic sum of several volumes bounded as above. For instance, to find the volume of a solid bounded by a convex closed surface we should circumscribe the solid by a cylinder whose gen- erators are parallel to "the z axis and then find the difference between two volumes like the preceding. Hence the formula (39) holds for any volume which lies between two parallel planes x = a and x = b (a < b) and which is bounded by any surface whatever, where A denotes the area of a section made by a plane parallel to the tv/o given planes. Let us suppose the interval (a, b) subdivided by the points a, x^ a;,, • • -, a;^_,, by and let Ao, Aj, • • •, Af, • • • be the areas of the sections made by the planes x = a, x = Xi, • • • , respectively. Then the definite integral j\ dx is the limit of the sum Ao(Xi - a) -f- Ai(x5, _ xi) + . . . + Ai_i(x,. - a;,._i) . • -. The geometrical meaningof this result is apparent. For A,_i (a;,. — a;,_i), for instance, represents the volume of a right cylinder whose base is the section of the given solid by the plane x = ar,_i and whose height is the distance between two consecutive sections. Hence the volume of the given solid is the limit of the sum of such infinitesimal cylin- ders. This fact is in conformity with the ordinary crude notion of volume. VI, $ 138] APPLICATinVS S86 If the value of the area A be known as a function of x, the toI- ume to be evaluated may be found by a single quadrature. As aa example let us try to find the volume of a portion of a •olid of evo- lution between two planes perpendicular to the axis of rerolutloiL Let this axis be the x axis and let « «/(x) be the equalkm of tkm generating curve in the xz plane. The seolioo made by a pUuie par- allel to the yz plane is a circle of radius f{x), Henoe the leqoired volume is given by the integral 'rrj^lf{x)Ydx. Again, let us try to find the volume of the portion of the ellipsoid ar* V* «• —-4- ^-4- —= 1 a« ^ 6« ^ c« ^ bounded by the two planes a; = a^o, « = X. The section made by a pla ne parall el to th e plane x = is an ellipse whose semiaxes are b Vl-xya=» and c Vl - x^/a\ Hence the volume sought is To find the total volume we should set atp= — a and JT as a, which gives the value ^Trabc. 138. Ruled surface. Phsmoidal formuU. When the 4rea A is aa iatsgnl function of the second degree in x, the volume nuy be expiwnd Tery dapiy in terms of the areas B and W of the bounding aectioiis, the ana 6 of the mmm section, and the distance h between the two boondiag ^«»f fAn f If the bmb section be the plane of yz^ we have r= r^"(te* + 2mx + n)dx = 2Z^ + »iio. J-a 8 But we also have A = 2a, 6 = n, B = Ufi -i- 2ma -^ n^ IT = (a* - tRUi -f n, whence n = h,a = A/2, 2ta* = B -f B" - 26. Theee equations kad to tke formaiA (40) r=:^[fi+B' + 461. which is called the prismoidai formula. This formula holds in particular for any soUd bduidtd by a two parallel planes, including m a qwcial caae the so-calkd prismokL* \ety = ax -\- p and z = to + 9 be the equations of a variable smiglil Bne, a, b, p, and q are continuous functions of a variable paruneter f whieb their initial values when t incre— ei frcHn l« to T. Tids stntsht Uae * A prismoid \h a solid bounded by aoy lei aud coutaiu all the vertices. — TaAits. 286 DOUBLE INTEGRALS [VI, § 139 a ruled surftwe, and the area of the section made by a plane parallel to the plane s = is giren, by S 94, by the integral A= r {ax-^p){b'x-hq')dt, wber« a', fc', C, d' denote the derivatives of a, 6, c, d with respect to t. These derivaUves may even be discontinuous for a finite number of values between to and r, which will be the case when the lateral boundary consists of portions of •everal ruled surfaces. The expression for A may be written in the form A = a;« f ab'dt + x f {aq' -\- pb')dt + f pq'dt, where the integrals on the right are evidently independent of x. Hence the formula (40) holds for the volume of the given solid. It is worthy of notice that the Mame/ormiUa also gives the volumes of most of the solids of elementary geometry. 189. Viviani's problem. Let C be a circle described with a radius OA {= R) of a given sphere as diameter, and let us try to find the volume of the portion of the sphere inside a circular cylinder whose right section is the circle C. Taking the origin at the center of the sphere, one fourth the required volume is given by the double integral - = / / Vi?2 - a;2 - y2 dxdy extended over a semicircle described on OA as diameter. Passing to polar coor- dinates p and w, the angle u varies from to 7t/2, and p from to E cos w. Hence we find Fto.81 — = - I (K» - i28 sm« w) dw = — I I • 4 8 Jo ^ ' 3 V2 3/ If this volume and the volume inside the cylinder which is symmetrical to this one with respect to the z axis be subtracted from the volume of the whole sphere, the remainder is 3 nR^ 8i?» /tt 2\ _ 3 V2 3 !\_ 16 R« Again, the area of the portion of the sur- face of the sphere inside the given cylinder is = 4 jy VrTp«Tg* dx dy . pwxiAqhf their valuM - z/t and - y/«, respectively, and passing to polar eottrdtnaUa, we find VI, f 140] APPLICATIONS f87 or = 4K»J''(1 - tin •-)di# = 4IP /- - iV Subtracting the area encloeed by the two ojlinden fiom Um wboto an* of tko sphere, the remainder ia 140. Evaluation of particular definite integrala. The tbeorOBf ettal^ lished above, in particular the theorem regarding diifereoiuiiUNi under the integral sign, sometimes enable us to evaluate oeitain defi- nite integrals without knowing the corresponding indefinite integrals We proceed to give a few examples. Setting the formula for differentiation under the integral sign giret Id ^ log(t-f g*) f xdx da" 1-ha* J. (l + «5)(l-fO* Breaking up this integrand into partial fractions, we find X 1 / J 4- *r _ g \ whence r xdx _ __ log(l4-a*) ■ tf ,nitani» X (l+ax)(l + x«) 2(1-Ha«) "^l+o* It follows that dA a ^ . log(l-f<t*) -r— =:r ; aTC tan a 4- 0.4 . -^ ' whence, observing that A vanishes when a as 0, we may write Integrating the first of these integrals hj parts, we finally find A =rarctanalog(l-h«*)- 2gg DOUBLE INTEGRALS [VI, §140 Again, consider ihe function x". This function is continuous when x'lies between and 1 and y between any two positive numbers a and h. Hence, by the general formula of § 123, J\ dx\ x^dy = j dy I x^dx. But ^ , hence the value of the right-hand side of the previous equation is On the other hand, we have ^ whence &,.UU^m. ^ ^..^ i^T^"^^=^^^(^i)- ^ '(^^ /r ^.r.'^.Mj^i^ In general, suppose that P(a5, y) and Q(x, 2^) are two functions \j^ -^ which satisfy the relation dPjdy = dQ/dx, and that iCo, a^i, 2/o> 2/1 are / u given constants. Then, by the general formula for integration under the integral sign, we shall have or [P(x, y,) - P(x, yo)]<«» = I [Q(a^i, y) - Q{x,, y)] c^y. Cauchy deduced the values of a large number of definite inte- grals from this formula. It is also closely and simply related to Green's theorem, of which it is essentially only a special case. For it may be derived by applying Green's theorem to the line integral jPdx -\- Qdy taken along the boundary of the rectangle formed l^ the lines jt = iCo, « = acj, y = yo, y = yi- In the following example the definite integral is evaluated by a •pecial derioe. The integral F(a) = / log (1 — 2a008x-^a^dx VI. i 140] APPLICATIONS f§§ has a finite value if \a\ is different from uni^v This fonelioo F(a) has the following properties. 1) F(- a) = F{a). For Fi^-a)^! :og(l4-2aooi« + «•)<{*, or, making the substitution x = ir — y^ ^(-«)=/ log(l-2acosy + a«)rfy«F(a). 2) F(a«) = 2F(a). For we may set 2F(a) = /^a) + F(-a), whence 2F(ar) = J [log(l - 2a cos X 4- a*) + log (1 4- 2a 008 z + a*)]*** = 1 log(l-2a^cos2z + a*)«ic. If we now make the substitution 2x = y, this becomas •2F(a) = iJ^ log(l-2a«c08y-f a*)rfy -f ^J^ log(l- 2a« cosy 4- a*)<fy. Making a second substitution y = 2tr — r in the last integral, w* find r log(l- 2^« cosy 4- a*)rfy=r log(l- 2a«oos« 4- a*)rf«, which leads to the formula From this result we have, successively, F(«) = |F(a«) = Jfr«*)=.. = iF(aO. If I a I is less than unity, <r** approaohes wsto as fi beoomet m^ouMi The same is true of F{it^), for the logwithm approtolMi Hence, if I or I < 1, we have F{a) = 0. iM DOUBLE INTEGRALS [VI, §141 If |a| U greater than unity, let us set a = 1/^. Then we find ''•)-x;-('-T^4)- = r log (1-2/3 cos x-f-)8')c?a;-7r log )8«, where |/5| is less than unity. Hence we have in this case F{a) = - TT log^ = TT log a^. Finally, it can be shown by the aid of Ex. 6, p. 205, that F{± 1) = ; hence F(a) is continuous for all values of a. 141. Approxinute value of logr(n + 1). A great variety of devices may be employed to find either the exact or at least an approximate value of a definite Integral. We proceed to give an example. We have, by definition, r(n + l) = r x'^er^dz. Jo The function x*c-* assiimes its maximum value n^e-" f or x = n. As x increases from zero to n, af*c-* increases from zero to n^c-" (n>0), and when x increases from n to + «, x*e-* decreases from n^e-" to zero. Likewise, the function n^er*e-^ increases from zero to n^e-" as t increases from — oo to zero, and decreaoes from n"e-'« to zero as t increases from zero to + oo. Hence, by the rabetitation (42) x"e-* = n"e-«e-«", the values of x and t correspond in such a way that as t increases from — oo to + 00, z increases from zero to + oo. It remains to calculate dx/dt. Taking the logarithmic derivative of each side of (42), we find dx_ 2tx dt~ x — n We have also, by (42), the equation <a = X -n -nlog (- j For simplicity let ua §et x = n + 2, and then develop log(l + z/n) by Taylor's theorem with a remainder after two terms. Substituting this expansion in the nXvm for (*, we find L~ 2n«(l+<?|fJ 2{n + dz)^ 9 liaa between zero and unity. From this we find, successively, r^.=K"^0 -[%!-<>-"]• VI, 5 142J APPUCAT10N8 291 whenoe, applying the fommJa for obaoga of Tariabl*, r(n + 1) = 2n-<- ^ f^^^^"^ + 2»-e- J** V^<U !)!<«. The flret integral U At for the lecond integral, though we cannot eTaloate it exactly, iIdm «• do not know 0, we can at least locate iu value between oeruin And t f^H i f^ all its elementA are negative between — oo and nro, and they an all pa^^ihm between zero and + oo. Moreover each of the integrmla /* , /^* la lea Ib absolute value than //* te-*'flM = 1/2. It follows that (43) Tin + 1) = v^ tfr^/vi + -^V where la lies between - 1 and + I. If n is very large, u/y/2n is very small. Hence, if we take r(n + 1) = n-e-^V^ni as an approximate value of r(n + !)« our error is reUtively small, tboogb tka actual error may be considerable. Taking the logarithm of each aide of (4S), w find the formula (44) log r(n + 1) = (n + 1) logn - n + 1 log(Sr) + 1. where e is very small when n is very large. N^ecting «, we hare an which is called the asymptx>tic value of logr(n + 1). TTiis formala la esting as giving us an idea of the order of magnitude of a f^oCorfal. 142. D'AIembert'8 theorem. The formula for integration under the sign applies to any function /(z, y) which is continnous in the rsetangit of fail*- gration. Hence, if two different results are obtained by two diflTerenl w alhods of integrating the function /(x, y), we may conclude that the foneCkai /(a, y) la discontinuous for at least one point in the field of integration. GaiMB dadncad from this fact an elegant demonstration of d*Alembert*s tbaoram. Let F{z) be an integral polynomial of degree m in s. W« shall aanaa for definiteness that all its coefficients are real. Replacing s by ^ooat* -f idaw), and separating the real and the imaginary parts, we have F(«) = P + <Q. where P = Aop'^ccmnua -{■ AifF^'^CMim - !)•# + ••• +il«, Q = i4o^sinm« + Ji^-» sln(m - 1)m + • •• -i-ila.t^alnw. If we set 1^ = arc tan (P/Q), we shall have ap~P«+Q«' d*» F«+^* and it is evident, without actually carrying out th« calculation, thai the derivative is of the form bpd^ (P« + W 292 DOUBLE INTEGRALS [VI, Exs. ^,„^ jf It ft conUnuoua function of p and «. This second derivative can only be dtoconlinuouB for values of p and w for which P and Q vanish simultaneously, that is to My, for the rooU of the equation F{z) = 0. Hence, if we can show that the two iDteg^rals an nnaqoal for a given value of iJ, we may conclude that the equation F{z) = has at least one root whose absolute value is less than B. But the second inte- gral is always zero, for Jf ^"i^Z; da, = f— T " ^^ » and bV/bp is a periodic function of w, of period 2;f. Calculating the first inte- gral in a similar manner, we find Jq 5p3w LS"Jp=o and it is easy to show that dV/d(a is of the form du> ~ AIp^^-\-"' le degree of the terms not written down is less than 2m in p, and where tha numerator contains no term which does not involve p. As p increases indefi- nitely, Uie right-hand side approaches — m. Hence R may be chosen so large that the value of dV/du, for p = iJ, is equal to — m + e, where c is less than m tat abeolute value. The integral /(,^"'(- m + c)dw is evidently negative, and the first of the integrals (45) cannot be zero. EXERCISES 1. At any point of the catenary defined in rectangular coordinates by the equation = |(ei + .-i) let OS draw the tangent and extend it until it meets the z axis at! a point T. Reroliing the whole figure about the x axis, find the difference between the areas deeeribed by the arc ilAf of the catenary, where A is the vertex of the catenary, and that deeeribed by the tangent MT {\) as a function of the abscissa of the poinl M, (8) as a function of the abscissa of the point T. [Licence^ Paris, 1889.] f. Using the usual system of trirectangular co5rdinates, let a ruled surface to formed as follows : The plane zOA revolves about the x axis, while the gen- WAtiBf line X>, which lies in this plane, makes with the z axis a constant angle whose taogent is X and ouU off on OA an intercept OC equal to \ad, where a If a fiteo leofth and # is the angle between the two planes zOx and zOA. VI, En.] EXERCISES 1) Find the Tolame of Um aoUd bounoed by iha nil«d fUifAot aod th» xOy, zOx, and zOA, where the angle $ between ihe laei two is Itm * ^n j 2) Find the area of the portion of the aorfaoe boonded bj the planet sOy xOx, zOA. [Liemf, Parit, July, 1881.] 3. Find the volumo of the solid boonded by the zy plana, iha eylladar 6^x^ + a'^y^ = a^6^, and the elliptic paraboloid wboae eqoatioa in rnHiiiiiilai coordinates is [Lkmee, Paiia, lltt.] 4. Find the area of the curvilinear quadrilateral bounded by the (bar focal conicfl of the family which are determined by giving X the values c«/8, S^/S, 4^/S, 6eV8, i [Lketiet, H—inyn, 18B6.] 5. Consider the curve y = V2(8inx ~oosz), where x and y are the rectangular co<)rdinatet of a point, and from it/i to 67r/4. Find : 1) the area between this curve and the z axis ; 2) the volume of the solid generated by revolving the eunre about the x axis ; 3) the lateral area of the same solid. [Lie€me€, Montpellter, 18ea.] 6. In an ordinary rectangular coordmaie plane let A and B be any two points on the y axis, and let .1 Mli be any curve Joining A and B whteh, tQfMter with the line AB, forms the boundary of a region AMBA wliOM araa la a pi<a- assigned quantity S. Find the value of the following daflnite latagnl over the curve A MB : f[<f>{v)e' -my]dx-{- [^'(jf)«« - Hdr . where m is a constant, and where the function ^(y), together with tti dMivttlvi <P'iy), is continuous. [Xieenet, Naney, IMi.] 7. By calculating the double integral / e-*v8inaxdifd:z in two different ways, show that, provided that a is not lero, "••^«d. = ±l r^*sin« 8. Find the area of the lateral surface of the portion of an sUlpsoid of revo- luUon or of an hyperboloid of revolntioo whkh Is boondsd bf two planes dicular to the axis of revolution. S94 DOUBLE INTEGRALS [VI, Exs. 9*. To ittd the area of an ellipsoid with three aneqaal axes. Half of the total ax«ft A ia given by the double integral '■■ffV^^W dxdy orer the interior of the ellipse b»x^ + a^y^ = a'^V^. Among the methods employed to reduce this double integral to elliptic integrals, one of the simplest, due to Catalan, consists in the transformation used in § 125. Denoting the integrand of the double integral by v, and letting v vary from 1 to + oo, it is eftqr to show that the double integral is equal to the limit, as I becomes infinite, ol the difference najb This expression is an undetermined form ; but we may write / v^dv V(— S)(— 8 c2\ L V f-tM and hence the limit considered above is readily seen to be irab ■{'-i)i-S)f dv W». If from the center of an ellipsoid whose semiaxes are a, 6, c a perpen- * T be let fall upon the tangent plane to the ellipsoid, the area of the surface la the locoa of the foot of the perpendicular is equal to the area of an •IHpiokl whose aeniiaxea are bc/a, ac/b, ab/c. [William Kobkkts, Journal de Liouville, Vol. XI, 1st series, p. 81.] VI, Ex. ] KXERCI8E8 11. Evaluate the double integral ut the ezpre extended over the interior of the triangle boonded bjr tbt itralflit Iteea y « a«, y = X, and z = X in two different ways, and tberebj wtabliah tb« formula /'dx/"(»-y)-/(y)dy= f ^^'^^^' m^- lilt deduce the relation f dzf dx " f Ax)dx = r (X - yrJ\y)uy. From this result deduce the relation In a similar manner derive the formula C'xdz f'xdx- f'xdx f'Ax)dz = __1_^ r V - l^)-/Ur)^. and verify these formulas by means of the law for diflamltatloB nadir tiM integral sign. CHAPTER VII MULTIPLE INTEGRALS INTEGRATION OF TOTAL DIFFERENTDLLS I. MULTIPLE INTEGRALS CHANGE OF VARIABLES 143. Triple integrals. Let F{Xj y, «) be a function of the three Tariables x, y, z which is continuous for all points M, whose rec- tangular coordinates are {x, y, z), in a finite region of space {E) bounded by one or more closed surfaces. Let this region be sub- divided into a number of subregions (^i), (eg), ••, (««), whose vol- umes are w„ V,, •••, v^, and let (^,-, 17,, Q be the coordinates of any point m^ of the subregion (e,). Then the sum (1) XK^i, -rn^^d^i 1=1 approaches a limit as the number of the subregions (e,) is increased indefinitely in such a way that the maximum diameter of each of them approaches zero. This limit is called the triple integral of the function F{xy y, z) extended throughout the region (E)^ and is represented by the symbol (2) / / / F{x,y,z)dxdydz. J J J(,E) The proof that this limit exists is practically a repetition of the proof giren above in the case of double integrals. Triple integrals arise in vai'ious problems of Mechanics, for instance in finding the mass or the center of gravity of a solid body. Suppose the region (E) filled with a heterogeneous sub- ftanoe, and let fi(x, y, z) be the density at any point, that is to say, the limit of the ratio of the mass inside an infinitesimal sphere about the point (x, y, «) as center to the volume of the sphere. If /oti and /aj are the maximum and the minimum value of /x in the subregion (e,), it \M evident that the mass inside that subregion lies between /ajV, and mv<; henoe it is equal to v./i(^„ ,;,, Q, where (^,, 7;., Q is a suitably choeen point of the subregion («,). It follows that the total 296 VII. $143] INTUODUCTION CHANGE OF VABIABLBS 297 mass is equal to thf^ triple integral ffffidxdpdm eitended Ihroofh- out the region (E). The evaluation of a triple integral maj be reduced to Ibe fo^ cessive evaluation of three simple integraU. Let ua auppoee first that the region (E) is a rectangular parallelepiped bounded by the six planes a; = jto, a; = ,Y, y = 5/0, y «= K, «■»«»,« — Z. h^ (K) he divided into smaller parallelopipeda by planet parallel to the three coordinate planer. The volume of one of the latter ia (^i — ar,-,) (y^ — y^_,) (zi — «|_i), and we have to find the limit ol the sum i L I where the point ((,iif Vutf Cm) i^ ^".v jxhmi 11 .r.' , parallelopiped. Let us evaluate tirst tliat ]- ^ .si... i. .i-. .-. from the column of elements bounded by the four plaoee x = Xf_,, r = Xi, y = y»-i, y = y», taking all the points (^^.,, rim, Cm) upon the straight line x bx^.,, y = yt-i' This column of parallelopipeds gives rise to the turn (x, - Xi_,)(y, - yt-,)[F(x,_,, y»_,, Ci)(«i -«•) + •••]» and, as in § 123, the ^'s may be chosen in such a way that the quantity inside the bracket will be equal to the simple integtal It only remains to find the limit of tiie sum • k But this limit is precisely the double integral <P{Xy y)dxdy extended over the rectangle formed by the linee « » «#, 9 « X, y — y(i>y= Y. Hence the triple integral is equal to C dxf ♦(X, y)dy, or, replafincr ^^/'.r. »/) by its value. (4) / ^^i "^J ''^'» y» *''*^- //• 208 MULTIPLE INTEGRALS [VII, § 144 The meaning of this symbol is perfectly obvious. During the first integration x and y are to be regarded as constants. The result will be a function of x and y, which is then to be integrated between the limits y« and K, X being regarded as a constant and y as a variable. The result of this second integration is a function of x alone, and the last step is the integration of this function between the limits Xq and X. There are evidently as many ways of performing this evaluation as there are permutations on three letters, that is, six. For instance, the triple integral is equivalent to ' dz\ dx f F(x, y, z)dy = j ^(z)dz, where ♦(«) denotes the double integral of F(x, y, z) extended over the rectangle formed by the lines x = Xq^ x = X, y = y^, y = Y. We might rediscover this formula by commencing with the part of the sum S which arises from the layer of parallelopipeds bounded by the two planes z = «^_i, z = Zi. Choosing the points (^, -q, ^) suitably, the part of S which arises from this layer is and the rest of the reasoning is similar to that above. 144. Let us now consider a region of space bounded in any manner whatever, and let us divide it into subregions such that any line parallel to a suitably chosen fixed line meets the surface which bounds any subregion in at most two points. We may evidently restrict ourselves without loss of generality to the case in which a line parallel to the z axis meets the surface in at most two points. The points upon the bounding surface project upon the xy plane into the points of a region A bounded by a closed contour C. To every point {x, y) inside C cor- respond two points on the bound- ing surface whose coordinates are We shall suppose that the functions Let us now Fio. 32 •i - ♦iC*! y) and «, s ^(x, y). ^1 and ^ are continuous inside C, and that <^i < </> VII, §144] INTRODUCTION CHANGE OP VARlAIUFs f^q divide the region under consideration i)y pUmcs panUiei to the ooflf- dinate planes. Some of the subdivisioni will be portiMii of ptnl- lelopipeds. The part of the sum (1) which ariiat from the ooIiudd of elements bounded by the four planee z « x^^i, « a x^, y ■■ y^.,, y = y^ is equal, by § 124, to the ezprestioD (^i - «<-i)(y* - y*-i) l^jT F(af,.„ y».„ M)dM -h ^\ where the absolute value of c^ may be made less than any promifncd number c by choosing the parallel planes sufficiently near together. The sum • k approaches zero as a limit, and the triple integral in question ij therefore equal to the double integral J JiA) *(x, y)dxdy I extended over the region {A ) bounded by the contour C, where the function 0(2;, y) is defined by the equation *(^, y) =J n^y y, ')dx. If a line parallel to the y axis meets the contour C in at meet points whose coordinates are y^= ^1(2) and y^^z ^t(x), respectiTelj, while X varies from x^ to 2:,, the triple integral may also be writteo in the form (6) f dx dy F{x,y,z)d*. The limits z^ and z^ depend upon both x and y, the limiti yi and |% are functions of x alone, and finally the limits X| and x, are cons t ants. We may invert the order of the integrations as for double inte- grals, but the limits are in general totally different for different orders of integration. Note. If ♦(x) be the function of x given by the double integral n') =f '^yj '^'C' y* *)*'« 800 MULTIPLE INTEGRALS [VII, § 145 extended over the section of the given region by a plane parallel to the yM plane whose abscissa is a:, the formula (5) may be written J I *^(x)dx. This is the result we should have obtained by starting with the layer of subregions bounded by the two planes x = Xi_Yj a; = x,. Choosing the points (^, 17, I) suitably, this layer contributes to the total sum the quantity ♦(«<-i)(aJ<-a;,_,). KxampU. Let as evaluate the triple integral fffz dx dy dz extended through- out that eighth of the sphere x^ + y'* + z^ = R^ which lies in the first octant. If we integrate first with regard to z, then with regard to y, and finally with regard to z, the limits are as follows : x and y being given, z may vary from zero to Vfi* — z* - y* ; X being given, y may vary from zero to VR^ — x^ ; and x itself may vary from zero to R. Hence the integral in question has the value J J J zdxdydz =J dx J dy J zdz, we find successively §X (ija - x« - y^)dy = [I (12^ - x^)y - ^ys]^ = §(^ " ^')^ and It merely remains to calculate the definite integral \S^{B^ - x^^dx, which, by the subetitution z = i2 cos 0, takes the form TT \ f ^R* sin* <f,d<f>. o Jo Hence the value of the given triple integral is, by § 116, 7tR*/lQ. 145. Change of variables. Let be formulffl of transformation which establish a one-to^ne corre- •pondenoe between the points of the region (E) and those of another region (^,). We shall think of m, v, and w as the rectangular coor- dioatet of a point with respect to another system of rectangular VII, §145] TNTRODUCTIOK nj wav np vaRUBLK*- ^01 coordinates, m ^'eiH»ral rlifTtrrut from the lir*t. If F(x, y, <) U a continuous function through' mu the ref^ion (E)f we shall alwavi hava (7) fff^F(x,y,z)dxdydx where the two integrals are extended throughout the reguma (K) and (Ei)j respectively. This is the formula for change of YariaUet in triple integrals. In order to show that the formula (7) always holds, we shall commence by remarking that if it holds for two or more particular transformations, it will hold also for the transformation obtained by carrying out these transformations in suoeession, bj the well-knowD properties of the functional determinant (S 29). If it is applicable to several regions of space, it is also applicable to the region obtained by combining them. We shall now proceed to show, as we did for double integrals, that the formula holds for a transformation which leaves all but one of the independent variables unchanged, — for example, for a transformation of the form (8) x = x', y = y', « = f(x', y*, «")■ We shall suppose that the two points ^f{xy y, z) and St(x\ y*, r^ an referred to the same system of rectangular axes, and that a paimllal to the z axis meets the surface which bounds the region {E) in at most two points. The formulae (8) establish a corre- spondence between this surface and another surface which bounds the region (K'). The cylinder circumscribed about the two sur- faces with its generators parallel to the z axis cuts the plane « = along a closed curve C. Every point w of the region A inside the contour C is the projection of two points my and rw, of the first surface, whose coordinates are «» and «,, respectively, and also of two points m[ and vi[ of the second surface, whose coordinates are «} and «i, respectively. Let us choose the notation in such a way that «i < «„ and «{ < «;. The formula (8) transform the point m^ into the point m|, or else into the point m^. To distinguish the two cases, we need merely consider the sign of df/dz'. If ^/^s' it ©0 rio.ss 802 MULTIPLE INTEGRALS [VII, § 146 positive, « increases with z\ and the points mx and m^ go into the points m{ and mi, respectively. On the other hand, if dy^/Jdz' is negative, x decreases as «' increases, and m^ and m, go into m[ and ml, respectively. In the previous case we shall have r F(x, y, z)dz= f F[x, y, ^(a;, y, «')] ^ «?«', whereas in the second case F(x, y, «)rf« = -J Fix, y, i/r(a;, y, ^O] ^, ^«'- In either case we may write (9) J\x,y,z)dz= £*Flx,y,^(x,y,z')2\^,dz'. If we now consider the double integrals of the two sides of this equation over the region A, the double integral of the left-hand side, 11 dxdy \ F(x, y, z)dz, J Ju) J*. is precisely the triple integral/// F(ic, y, z) dx dydz extended through- out the region {E). Likewise, the double integral of the right-hand side of (9) is equal to the triple integral of n^\ y\ K^\ y\ ^')] extended throughout the region (£:'), which readily follows when X and y are replaced by a;' and y', respectively. Hence we have in this particular case J J J{K) F(aj, y, z) dx dy dz J J J(E') ^[^',y',^(a;',y',«')] dz' dx'dy'dz'. But in this case the determinant D(x, y, z)/D(x', y', «') reduces to 9if/dM\ Hence the formula (7) holds for the transformation (8). Again, the general formula (7) holds for a transformation of the type (10) m »/(x', y', •') , y = ^(x', y', z'), z^ k\ VU,514fl] INTRODUCTION CHANGE OF VARIABLES where the variable z remaina unchaoged. We aball tuppoto ^ Ki l the formulae (10) establish a one-toone oorreapondenoe betwtM the points of two regions (E) and (K'), and in particular thai the sections R and R' made in (E) and (^*), respectivelj, bj aaj plane parallel to tlie xy plane correspond in a one-to-one manner. Then by the formulae for transformation of double tntegrmls W9 shall have (11) / / ^(«» y,z)dxdy The two members of this equation are functions of the rariable z = z' alone. Integrating both sides again l^tween the limits S| and «a, between which « can vary in the region (E), we find Um formula fffF{x,y,z)dxdydM J J JiE) (12) But in this case D{t, y, «)//>(x', y', z) = i\x, y)/D(x', y^. Heooa the formula (7) holds for the transformation (10) also. We shall now show that any change of variables whafeem (13) a;=/(x„yi, «i), y = <^(a:i, yi, «,), s « ^x,, y,, «,) may be o'btained by a combination of the preceding transformations. For, let us set x' = x^, y' = y^ z' s z. Then the last equation of (13) may be written z' = ^(x'f y\ «,), whence «, = ir(x*, y*, «^ Hence the equations (13) may be replaced bv the six e<]uatlons (14) X =/[x', y\ 7r(x', y\ *')], y = ^[x', y', - (16) x' = aH, y' = yi, «' = ^^,-^1, yx, «i The general formula (7) holds, as we have seen, for each of the transformations (14) and (15). Henoe it holds for the transforma- tion (13) also. We might have replaced the general transformatinn (13>. .^^ th** reader can easily show, by a seciufan-** nf thnv trannform*tion!i *^{ the type (8). 804 MULTIPLE INTEGRALS [VII, §146 146. Element of volume. Setting F(aj, y, «) =1 in the formula (7), we find J J JiE) J J JiE^\ Djx, y, z) D(u, V, w) dudvdw. The left-hand side of this equation is the volume of the region (.E:). Applying the law of the mean to the integral on the right, we find the relation (16) V = V, DLfy <^> «A) /)(«, V, w) (f,»»,o' where Kj is the volume of {EC), and ^, -q, ^ are the coordinates of some point in {Ex). This formula is exactly analogous to formula (17), Chapter VI. It shows that the functional determinant is the limit of the ratio of two corresponding infinitesimal volumes. If one of the variables m, v, w in (6) be assigned a constant value, while the others are allowed to vary, we obtain three families of surfaces, u = const., v = const., w = const., by means of which the region (E) may be divided into subregions analogous to the paral- lelopipeds used above, each of which is bounded by six curved faces. The volume of one of these subregions bounded by the surfaces (tt), (u 4- du), (r), (v -f dv), (w), (w -f dw) is, by (16), ^V=\\^^^^\i.Adudvdw, where rfw, rfv, and dtv are positive increments, and where c is infini- tesimal with du, dv, and dw. The term e du dv dw may be neglected, M has been explained several times (§ 128). The product (17) rfK = P^-^"»'^) du dv dw \D{Uy V, w) is the principal part of the infinitesimal AF, and is called the element of volume in the system of curvilinear coordinates (u, v, w). Let d** be the square of the linear element in the same system of coordinates. Then, from (6), whence, squaring and adding, we find X ^Hxdu^-¥Utdf)^-\-lUdw^-\-2F^dvdw+2F^dudw+2F,,dudv, VII. J 146] INTKODUCTION CHANGE OF VARIABLES S05 the notation employed being (19) (".-^©■' "sm: «..sm: 'Bxy F» ^ bu cv where tlie symbol S means, as usual, that x is to be replaeed by y and « successively and the resulting terms then added. The formula fcr dV is easily deduced from this formula for d^. For, squaring the functional determinant by the usual rule, we find Af, whence the element of volume is equal to "VMdu dv dw. Let us consider in particular the very important caae in whieh the coordinate surfaces (u), {v)y (ta) form a triply orthogonal Bjtt«m, that is to say, in which the three surfaces which pass through any point in space intersect in pairs at right angles. The tnngentt to the three curves in which the surfaces intersect in pairs form a tri- rectangular trihedron. It follows that we must have F} sa 0, F, ■■ 0, F, = ; and these conditions are also sufficient The formubs for rfl' and ds^ then take the simple forms dx dj. dx t du du du //l F, Ft dx hi dv dz dv dv = Ft //. Fi dx d_y dx Ft ^1 B. dw dw dw du dv dw. (20) ds^ = Hi du' 4- //, dv* + //, dw*, dV = >///,//,//. These formulae may also be derived from certain considerations of infinitesimal geometry. Let us suppose du^ dv, and dw very small, and let us substitute in place of the small subregion defined ftbore a small parallelopiped with plane faoes. Neglecting infinit esim a l s of higher order, the three adjacent edges of the parallelopiped may be taken to be yJlT^du, y^/Tf^dv, and y/W^dw, respectively. Tbe for- mulae (20) express the fact that the linear element and the elemeot of volume are equal to the diagon al and the volume of this parallelo- piped, respectively. The area v/i/j //, du dv of one of the faoes repre- sents in a similar manner the element of area of the surface (w). As an example consider the transformation to polar coordinates (21) x = psindco8</>, y = p8intf8in4, s«peos#, 306 MULTIPLE INTEGRALS [VII, § 146 where p denotes the distance of the point M(Xf y, z) from the origin, B the angle between OM and the positive ;;; axis, and <^ the angle which the projection of OM on the xij plane makes with the positive X axis. In order to reach all points in space, it is sufficient to let p vary from zero to 4- oo, ^ from zero to tt, and <^ from zero to 27r. From (21) we find (22) ds'' whence dp^ -^ p^dO'' ^ p'^ sm''ed4>'^, (23) dV = p"^ sine dp ddd<i>. These formulae may be derived without any calculation, however. The three families of surfaces (p), (d), (<^) are concentric spheres about the origin, cones of revolution about the z axis with their vertices at the origin, and planes through the z axis, respectively. These surfaces evidently form a triply orthogonal system, and the dimen- sions of the elementary subregion are seen from the figure to be dp, p d$, p sin e d(f> ; the formulae (22) and (23) now follow immediately. To calculate in terms of the va- riables p, 6, and </> a triple integral extended throughout a region bounded by a closed surface S, which contains the origin and which is met in at most one point by a radius vector through the origin, p should be allowed to vary from zero to R, where R =/($, <f>) is the equation of the surface ; $ from zero to Fio. 34 tt: and <^ from zero to 27r. For example, the volume of such a surface is X2.T ^n pR d<t> j del p^ sine dp. The first integration can always be performed, and we may write OocMional use is made of cylindrical coordinates r, o), and z defined by the equations x = r cos «, y = r sin cu, « = z. It is evident that dV =si r dit) dr dz , VII. §147] INTRODUCTION CHANGE OP YARUBLES ttfj 147. Elliptic coii^dinatet. The mutmom reprsMBled by ut« e4uatK]a X* V* «• X-o X-6 X-c %rhere X Ib a rariable parameter and a > 6 > c> 0, fom a teailj o( conies. Through every point in space there pa« three anrfaees of this faadljr,—^ an ellipsoid, a parted hyperboloid, and an unparted hjperbolold. For the eq«»> tion (24) always has one root Xi which liee betwee n b and c, another rooi X« between a and 6, and a third root Xt greater than a. Theee three roou X|, X«, Xg are called ihe elliptic coordinaUa of the point whoee rectangular frfrftrdinaif are (z, y, z). Any two surfaces of the family int«reeet at right angles: if X he given the values Xi and X3, for instance, in (24), and the rseoltlng eqoationa be snb- tracted, a division by Xi — Xt glyes ^ ' (Xi-a)(X,-a) (Xi-6)(X,-6) (Xj - c)(X, - c) which shows that the two surfaces (Xi) and (X9) are orthogcmal. In order to obtain z, y, and z as functions of Xi , Xf , Xa , we maj note that the relation (X _ a)(X - 6)(X - c) - x^{\ - b){\ _ c) - y«(X - c)(X - a) - f«(X - «)(X - b) = (X - X|)(X - X,)(X - X.) is identically satisfied. Setting X = a, X = 6, X = e, i ucn ee i irely, in tide eqQ»- tion, we obtain the values r,,_ (X«-a)(a~X,)(a~X,) I (a-b)(a-c) • (X. - 6)(X, - b){b - XQ (26) V (a-6)(6-c) ^ ^ (X, - c)( X,-c)<Xi~c) (a - c)(6 - c) whence, taking the logarithmic deriTatiree, S \X, - o X, - « X, - «/ ,/ dX, (IX. (ft, \ *-iii:r^e+i:r^e+xr^«r Forming the sum of the squares, the terms In (iX,tfX«, dXfAa, AeA| »■* dto appear by means of (26) and similar relaUons. H^ne* the ooeiBeioiil of tfX, li ^' " 4 L(xr^« "*" (xr=^ "" ^ - «>• J ' or, replacing x, y, t by their valnee and simplifying, (X« - XO(Xt - X|) (27) Ml ^2 4 (Xi - a)(X, - 6)(X, - e) $08 MULTIPLE INTEGRALS [VII, §148 eoeflleients lit and Jf| of <ix| and dxj, respectively, may be obtained from ion by cyclic permutation of the letters. The element of volume is thtref ore VMiMt Mt d\x dX, dX. . 141. Dirichlet't integrals. Consider the triple integral xPy««'"(l — X — y — z)'dxdydz fff' throaghout the interior of the tetrahedron formed by the four planes ssO, ir = 0, « = 0, x4-y + «=l. Let us set x + y-\-z = i, y + z = ^v, z = ^r]^, where (, If, f are three new variables. These formulae may be written in the form y -\- z ^ z x + y + z y + z and the inverse transformation is x = ^{i-v), y = ^v{i-^), z = hj:. When X, y, and z are all positive and x + y + zis less than unity, ^, rj, and f all lie between zero and unity. Conversely, if ^, ?;, and f all lie between zero and unity, x, y, and z are all positive and x-^y + z is less than unity. The tetra- hedron therefore goes over into a cube. In order to calculate the functional determinant, let us introduce the auxiliary transformation JT = f , r = fi;, Z = ^r^f , which gives x = X-Y, y = Y-Zy t = Z. Hence the functional determinant has the value D(x,y, z) ^ D(x, y, z) J)(X, F, Z) ^ D(^7,, f) i)(X, r, Z)" D(|, ^, f) ^''' and the given triple integral becomes The integrand le the product of a function of {, a function of rj, and a func- tion of f. Hence the triple integral may be written in the form or, introducing r functions (see (33), p. 280), Tip + q-^r-^ B)r{i +i) ^^ r(g -f r -t- 2)r(p +1) r(r + i)r(g4-i) r(p + y + r + « + 4) r(p + g + r + 3) ^ r(94-r + 2) ' CanotUnf the common factors, the value of the given triple integral is finally found to he (88) r(p4-l)r(g4-l)r(r.fl)r(a+l) r(p + 5 + r + « + 4) VII. §149] INTKODUCTION CHAMGE OF VAKIABLES 809 149. Green'i thwutm.* A formula entirely MuOogoiu lo (lo^, | IM, m»y te derived for triple integralB. Let us arat coiulder a etoaed torfiM 8 wkUk to met in at most two poinu by a parallel to the < axii, and a funeUoo A(s, y, g) which, together with dH/^x, Is continuous thrauglkout the iniarior ol tbianrteoa. All the points of the surface 8 project into poinu of a ragloD il ollbe ay r^^ which iH bounded by a closed contour C. To every point of A »— Mt C eon*- spond two pointo of 8 whose coordinates are <i = ^i(z, y) and sg a ^(x« y). The surface S is thuM divided into two distinct fxiriions ^i aod iBg. We ««*i ^« suppose that Zi is less than Zt. Let us now consider the triple integral /// — ozdyds £z taken throughout the region bounded by the closed surface 5. A .-^ ,,, ,, tion may be performed with regard to t between the limita Mt and ft (f 144), which gives K(x, y, zt) - K{x, y, zi). The given triple Integral to thwifwi equal to the double integral //[«(x, y, Zt) - «(x, y, tOJdedy over the region A. But the double integral ffR(x, y, c«)dzdy to equal to the surface integral (§ 13o) ff R{x,y,z)dx(fy taken over the upper side of the surface St . Likewise, the R{x, y, zi) with its sign changed is the surface integral ff R(x,y,t)dxd^ 8i . Adding these tw taken over the lower side of 8i . Adding these two Integrato, we may where the surface integral is to be extended over the whole mimriar of the face S. By the methods already used several times in similar caaee tl be extended to the case of a region bounded hy ^warho^oiMaj Again, permuting the letters z, y, and t, we obtain the aaalogoat /// /// ~dxdvdz=ff P(x, y, f)d|rdi, ^^dzdy<U=ff Q(x, y, s)dt4a. dy J JfS) • Occasionally called Osfro^rodciby'* theorem. The t h eo t e w of f IS to i called Iiiema7m'.H theorem. But the title Oreen*» CAsorem to SMwe etoariy and seems to be the more fitting. See Mnqf, dtt Mmtk. IRss., II, A, 7. h aad e.— Trams. ^^ MULTIPLE INTEGRALS [VII, §150 Adding thm& three formula, we finally find the general Green's theorem for triple integrals : (2D) = ff P(x, y, z)dydz + Q{x, y, z)dzdx + 12(x, y, 2:)(icdy, J J{5) when the surface integrals are to be taken, as before, over the exterior of the bounding surface. If, for example, we set P = x, Q = R = or Q = y, P = R = or R = z, P = Q = , it is evident that the volume of the solid bounded by S is equal to Any one of the surface integrals (290 IL"''^'' /i/"'^' /^^'"'^ 150. Multiple integrals. The purely analytical definitions which have been gi?en for double and triple integrals may be extended to any number of vari- ables. We shall restrict ourselves to a sketch of the general process. Let zi , Xt , • • • , x„ be n independent variables. We shall say for brevity that a system of values x}, x§, • • • , xj of these variables represents a point in space of n dimensions. Any equation F(xi, X2 , • • • , x„) = 0, whose first member is a continuous function, will be said to represent a surface ; and if F is of the first degree, the equation will be said to represent a plane. Let us consider the totality of all points whose codrdinates satisfy certain inequalities of the form (80) ^.(xi,««, •••,x„)<0, i = l, 2, ..., A:. We shall say that the totality of these points forms a domain D in space of n dimensions. If for all the points of this domain the absolute value of each of the cotedinates x, is less than a fixed number, we shall say that the domain B is finite. If the inequalities which define D are of the form (81) «;<«i<xi, x;<x,<x5, ..., x;<x„<xl, we shall call the domain a priamoid, and we shall say that the n positive quan- tities x} - a^ are the dimensions of this prismoid. Finally, we shall say that a point of the domain D lies on the frontier of the domain if at least one of the functions ft in (80) vanishes at that point. Now let D be a finite domain, and let /(xi , Xa , • • • , Xn) be a function which b continuous in that domain. Suppose D divided into subdomains by planes pAfallei to the planes x, = (f = 1, 2, •• • , n), and consider any one of the pris- moida determined by these planes which lies entirely inside the domain D. JM Asi , Axtt " - 1 Ax» be the dimensions of this prismoid, and let fi , fa , • • • , t, bt tba coordinates of some point of the prismoid. Then the sum (tl) fl = 2/({„ {,, . . ., („) AxiAx,. . . Ax«, formad for all the prlsmoids which lie entirely inside the domain D, approaches a limit / as the number of the prlsmoids is increased indefinitely in such a way VII, §150] INTRODUCTION CHANGK OF VAUIABLES 311 that all of the dimenaiooi of each of ibem np\n limit / the n-tuple integral of/(Xi, 2«, • • •, z.) ta. denote it by the symbol /=// • //(xi. X,. .. .. x,)d£,(ljr. The evaluation of an n-tuple integral may be reduced to the etaJoalloci ci n successive simple integrals. In order to show this In general, we need only show that if it is true for an (n - l)-tuple integral, it will aleo be Ime for an »-tuple integral. For this purpose let us consider any point (Xi, St«* *-• <■) of D. Discarding the variable x« for the moment, the point (Zi , Zt t "*•'«- 1) c^ dently describes a domain 1/ in space of (a - 1) dlmendona. We shall sappoM that to any point (xi, xt, • •, z«-i) inHide of IX there oorreepond Juet two points on the frontier of D, whoee coordinates are (zi, Xt, • • •, z..! ; z^J^ and (xi , za , • • • , Xm-1 ; z^/^), where the coordinates z^^ and z^" are oontinuoue fan6> tions of the n - 1 variables Zi , z^ , • • • , z.-i inside the domain IX. If this eeo- dition were not satisfied, we should divide the domain D into domains i that the condition would be met by each of the partial domains. Let consider the columu of prismoids of the domain D which correspond to the same point (zi, z^, • • •, z«-i). It is eaey to show, as we did in the similar eaae treated in § 124, that the part of S which arlaet from thia column of prtaHwida ii AziAxt ••Az,_irr '/(zi, %,..., jgdz, + €j. where |e| may be made smaller than any positive number wbaterer by cIkk»> ing the quantities Az, sufficiently smalL If we now set (88) ♦(zi,z,,...,z,_,)=j^,^/lzi,a^,...,*jdz.. it is clear that the integral I will be equal to the limit of the 2*(zi, z«, •••, z,_i)AzjAZf.Az,_i, that is, to the (n - l)-tuple integral (84) 1 = JJJ •.J*(zi,z,,. •,z._i)dzi...dj|,-i, in the domain IT. The law having been ■oppoeed to bold for an (• - IHaple integral, it is evident, by mathematical indadkm, that It bokla in geoeraL We might have proceeded difterenUy. Consider the loulity of points (xu Xt, . . •, Xn) for which the coordinate z, has a fixed valne. Tbeo the point (zi, zi, . • •, Xn-i) describes a domain i in apnea of (n - 1) dIavMinm, and it is easy to show that the Wrtuple integral / ia alio equal to the ■ipiaaliia ■ (86) I^f^eiz.)dx.. where 0{Xn) is the (n - l)-tuple integral ///• • f/dtt • • • da. - 1 ezlMided throogb. out the domain «. Whatever be the method of carrying oat Uie proesM, the Umlta for the various integrations depend upon the naiiire of Ibe donate A ■■* 312 MULTIPLE INTEGRALS [VII, §160 Tary to general for different orders of integration. An exception exista in case D is a priamoid defined by inequalities of the form a?^«i<Xi, •., xJ<x,<X., .... The multiple Integral ia then of the form and the oitler in which the integrations are performed may be permuted in any way whaterer without altering the limits which correspond to each of the variables. The formula for change of variables also may be extended to n-tuple integrals. Let (36) x, = 0,(xi',a^, •••,x;), i = l, 2, ..-, n, be formuliBof transformation which establish a one-to-one correspondence between the points (x'l, Xa » • • • , Xn) of a domain 1/ and the points (xi, Xa, • • • , x„) of a domain D. Then we shall have ff"'f ^(^i'^'*"'^«)^i""^» (87) J J J(L F{<f>i » • • • » 0«) -D(0i, •••,0«) I>(xi, ••«, x;) dxi- • ■ dXn . The proof is similar to that given in analogous cases above. A sketch of the argument is all that we shall attempt here. 1) If (37) holds for each of two transformations, it also holds for the trans- formation obtained by carrying out the two in succession. 2) Any change of variables may be obtained by combining two transforma- tions of the following types : (88) «i = xi, x, = x$, •.., x„_i = x;_i, x„ = 0„(x{, x^, ..-, x;), (89) «i = fi(xi,..-,x;), ..., Xn-i = i/'„-i(xl, .-.jx;), x„ = xA. 8) The formula (37) holds for a transformation of the type (38), since the given yi-tuple integral may be written in the form (34). It also holds for any transformation of the form (39), by the second form (35) in which the multiple totegral may be written. These conclusions are based on the assumption that (87) holds for an (n — l)-tuple integral. The usual reasoning by mathematical toduction establishes the formula in general. As an example let us try to evaluate the definite integral ^ =//• • -/ac^xj' . . . x;-(l - xi - xa Xn)^dxidX2. . .dx„, where ai, at, ■ • , or., /3 are certain positive constants, and the integral is to be CTt ended throughout the domain D defined by the inequalities 0<x,, 0<Zs, ..., 0<x,., Xi + X2 + .-. + »,<l. The transformation Vll, §161] TOTAL DIFFEUENTlAiJi $|| carries I> into a new domam if ueiineu by loe ineqiuuiuet 0<(i<l, 0^|,<1, ..., 0<|.^1, and it is easy to show as in f 148 that the Tains of th« •■^■w»Hi^af rtstwlnm to ^git«tt "s a^i) _>•-!>■ -t . iAJn^ (tt •••, Km) The new integrand is therefore of the form and the given integral may be expressed, as before, in terms of r fnoctioBS i (40) I = ^'^^ 4-l)r(a , H-l)...r(a,.H)r(^^l) r(ai + a, + • • • + a. + /I + m + 1) n. INTEGKATION OF TOTAL DIFFERENTIALS 151. General method. Let /'(x, y) aiid Q(x, y) be two functions of the two independent variables x aiid y. Then the expreeeioo Pete -{- Qdy is not in general the total differential of a single function of the two variables x and y. For we have seen that the equation (41) du^Pdx^-Qdy is equivalent to the two distinct equations (42) t='P(^.y). |^=«(''y)- Differentiating the first of these equations with reepeot to y and tlie second with respect to x, it appears that m(z, y) must satisfy eaoh of the equations c^u ^ dP(x,y) Bl^u ^ dQ(x,y) dxdy" dy ' dydx"^ dx A necessary condition that a function «(«, y) should exist which satisfies these requirements is that the equation (''3) ay to should be identically satisfied. This condition is also st^ffleieni. For there exist an in6nit» number of functions w(x, y) for which the first of equations (43) is satisfied. All these functions are given by the formula u^j Pix,y)dX'^Y, 814 MULTIPLE INTEGRALS [VII, §151 where oiiiq is an arbitrary constant and Y is an arbitrary function of y. In order that this function u{x^ y) should satisfy the equation (41), it is necessary and suflScient that its partial derivative with respect to X should be equal to Q(Xj y), that is, that the equation should be satisfied. But by the assumed relation (43) we have whence the preceding relation reduces to dY ^ = Q(^o,y). The right-hand side of this equation is independent of x. Hence there are an infinite number of functions of y which satisfy the equation, and they are all given by the formula Y=\ Q(x,,y)dy^C, where y© is an arbitrary value of y, and C is an arbitrary constant. It follows that there are an infinite number of functions u{x, y) which satisfy the equation (41). They are all given by the formula (44) u=\ P(x, y)dx-h f Q(xo, y)dy + C, and differ from each other only by the additive constant C. Consider, for example, the pair of functions a; + my ^ _ y - ma; x' + y'' ^-^M^' which satisfy the condition (43). Setting a^o = and y^ = 1, the formula for u gives Jo x^ + y^ J^ y » whence, performing the indicated integrations, we find = 2 P*^»(** + y*)]* + ^'^ r^c tan -T 4- log y 4- e, U =s or, simplifying, 1 tt = 2 lo«(»* ■fy*) + marctan--hC, VII, 5 151] TOTAL DIFFERENTIALS $16 The preceding method may be extended to any number of inde- pendent variables. We shall give the reasoning for tbree Tariablet. Let Py Qj and R be three functions of x^ y^ and z. Then the total differential equation (46) du = luU -j- Udi/ -}- It dz is equivalent to the three distinct equatiouw Calculating the three derivatives d^u/dxdy^ dl*u/dybUfd^u/dzbx in two different ways, we find the three following equationi at naoea- sary conditions for the existence of the function m; (47) ^ = ^, £^ = ^, i^^^. dy dx* Bk dy* dx du' Conversely, let us suppose these equations satisfied. Then, bv tiie first, there exist an infinite number of functions u{Xy y, t) whose partial derivatives with respect to x and y are equal to P and Q, respectively, and they are all given by the formula t*=/ P{x,y,z)dx^\ Q{Xf,,y,z)dy'k'Z, where Z denotes an arbitrary function of z. In order that the derira- tive du/dz should be equal to i?, it is necessary and sufficient that the equation should be satisfied. Making use of the relations (47), whieh wet* assumed to hold, this condition reduces to the equation dZ R(x, y, z) - /e(Xo, y, z) -f R(x^, y, z) - R(x., y., «) + ^ = i?(x, y, s) , It follows that an infinite number of functions u(x, y, z) exist which satisfy the equation (46). They are all giren by the fonnula (48) u= f P(x, y,z)dx-h f Qix^y y. r^ du + ( R(x., y.. z^ dx + r. where Xq, i/,,, z^, are three arbitrary numerical values, and r i> ;in arbitrary constant. 310 MULTIPLE INTEGRALS [VU,§162 152. The Integral /^'J^Pdx-fQdy. The same subject may be treated from a different "point of view, which gives deeper insight into the question and leads to new results. Let P(x, y) and Q{xy y) be two functions which, together with their first derivatives, ai-e continuous in a region A bounded by a single closed contour C. It may happen that the region A embraces the whole plane, in which case tlie contour C would be supposed to have receded to infinity. The line integral / Pdx + Qdy taken along any path D which lies in A will depend in general upon the path of integration. Let us first try to find the conditions under which this integral depends only upon the coordinates of the extremi- ties (jTg, y^ and (ajj, y^) of the path. Let M and N be any two points of region Ay and let L and V be any two paths which connect these two points without intersecting each other between the extremities. Taken together they form a closed contour. In order that the values of the line integral taken along these two paths L and V should be equal, it is evidently necessary and sufficient that the integral taken around the closed contour formed by the two curves, proceeding always in the same sense, should be zero. Hence the question at issue is exactly equivalent to the following : What are the conditions under which the line integral I Pdx-{-Qdy taken around any closed contour whatever which lies in the region A should vanish ? The answer to this question is an immediate result of Green's theorem : (49) i^-^^<^^y=Jfl^i-'iy^y> where C is any closed contour which lies in .4, and where the double integral is to be extended over the whole interior of C. It is clear that if the functions P and Q satisfy the equation the line integral on the left will always vanish. This condition is necessary. For, if cP/dy - dQ/dx were not identically zero VII, §152] TOTAL DIFFEUEN'TrAW gl7 in the region .4, since it is a continuuu^ luuction, it would turelj be possible to find a region a so small that its sign would be oonstant inside of a. But in that case the line integral taken around the boundary of a would not be zero, by (49). If the condition (43*) is identically satisfied, the Talnei of the integral taken along two paths L and /.' between the same two points M and N are equal provided the two paths do not intarteei between M and N. It is easy to see that the same thing is true even when the two paths intersect any number of times between Mt and ^V. For in that case it would be necessary only to oompare the values of the integral taken along the paths L and L* with its value taken along a third path L'\ which intersects neither of the preceding except at M and N. Let us now suppose that one of the extremities of the path of integration is a fixed point (xo, y^y while the other extremity is a variable point (x, y) oi A. Then the integral I Pdx + Qdy <'•• »•> taken along an arbitrary path depends only upon the oo6rdinates (x, y) of the vaiiable extremity. The partial derivatiTee of this function are precisely P(x, y) and Q(x, y). For example, we have for we may suppose that the path of integration goes from (x^, f^) to (x, y),.and then from (x, y) to (x + Ax, y) along a line parallel to the X axis, along which dy = 0. Applying the law of the mean, we may write Ax Taking the limit when Ax approaches »ro, this gives P^»P, Similarly, F, = Q. The line integral F(x, y), therefore, satisfies the total differential equation (41), and the general integral of this equation is given by adding to F(x, y) an arbitrary constant This new formula is more general than the formula (44) in that the path of integration is still arbitrary. It is easy to deduce (44) from the new form. To avoid ambiguity, let (x«, y.) and (x,. y,) be the co5rdinates of the two extremities, and let the path of int4^r»- tion be the two straight lines x = x^, y = yi. Along the former, 818 MULTIPLE INTEGRALS [Vll, §153 X =1 Xo» dx = Of and y varies from ijo to yi. Along the second, y = yi, rfy = 0, and X varies from Xq to a^i. Hence the integral (50) is equal to J' 'Q(xoyy)dy+f P(x,yi)dx, which differs from (44) only in notation. But it might be more advantageous to consider another path of integration. Let x = /(^), y = <f>(t) be the equations of a curve joining (xo, y©) and (xi, yi), and let t be supposed to vary con- tinuously from to to ti as the point (x, y) describes the curve between its two extremities. Then we shall have r '"'^Pdx + Qdy^f \p(x, y)f'(t) + Q{x, y) ^\t)-] dt, where there remains but a single quadrature. If the path be a straight line, for example, we should set x = Xq-}- t(xi — Xq)j y = yo + t(yi — yo), and we should let t vary from to 1. Conversely, if a particular integral ^(x, y) of the equation (41) be known, the line integral is given by the formula I Pdx + Qdy=:^(x,y)-^(xQ,yQ), which is analogous to the equation (6) of Chapter IV. 153. Periods. More general cases may be investigated. In the first place, Green's theorem applies to regions bounded by several contours. Let us consider for definiteness a region A bounded by an exterior contour C and two contours C and C" which lie inside the first (Fig. 35). Let P and Q be two functions which, together with their first derivatives, are continuous in this region. (The regions inside the contours C and C" should not be considered as parts of the region A^ and no hypothesis whatever is made regarding P and Q inside these regions.) Let the contours C and C" be joined to the contour C by trans- ▼eraals ab and ed. We thus obtain a closed contour abmcdndcpbaqay or r, which may be described at one stroke. Applying Green's theorem to the region bounded by this contour, the line integrals VU,J153] TOTAL DIFFERENTIALS 810 whicli arise from the traDBversaU ab and cd eanoel out, tioM » ^h of them is described twioe iu opposite direotiona. It follows that where the line integral is to be taken along the whole boondarj of the region A, Le. along the three contours C, C, and C'\ in tho mmum indicated by the arrows, respectively, theso being tuoh that tha region A always lies on the left If the functions P and Q satisfy the relation dQ/dx » BP/^ in the region .1, the double integral vanishea, and we may write the resulting relation in the form (51) r Pdx + Qdi/= f Pdx-i- Qdy -f- / Prfx + Wy, JiC) J{V) J{C"} where each of the line integrals is to be taken in the aenae dcaig- uated above. Let us now return to the region A bounded by a single eontoor C, and let P and Q be two functions which satisfy the eqoatioo dP/dy = dQ/dxy and which, together with their first deriTatiTea, are continuous except at a finite number of points of Af at which at least one of the functions P or Q is discontinuous. We shall suppose for definiteness that there are three points of discontinuity a, by c in .4. Let us surround each of these points by a small circle, and then join each of these circles to the contour C by a cross cut (Fig. 36). Then the integral jPdx-\-Qdy taken from a fixed point (xo, yo) to a variable point (ar, y) y,Q. as along a curve which does not croes any of these cuts has a definite value at every point For the ooaUwr C, the circles and the cuts form a single contour which may be deeeribed at one stroke, just as in the case disouased aboTO. We iliall eall such a path direct, and shall denote the ral oe of t he line intef^ taken along it from 3fo(j-o, y*) to3f(a:, y) by F{x, y). We shall call the path composed of the straight line from U^ u> a point a\ whose distance from a is infinites im al, the eironMfwsMa of the circle of ra<liu8 aa' about a, and the straight line a*M%t a l»^ circuit. The line integral jPdx + Qdy taken along a loop-emiil 820 MULTIPLE INTEGRALS [VII, §153 reduoes to the line integral taken along the circumference of the circle. This latter integral is not zero, in general, if one of the functions P or Q is infinite at the point a, but it is independent of the radius of the circle. It is a certain constant ± A, the double sign corresponding to the two senses in which the circumference may be described. Similarly, we shall denote by ± B and ± C the values of the integral taken along loop-circuits drawn about the two singular points h and c, respectively. Any path whatever joining Mq and M may now be reduced to a combination of loop-circuits followed by a direct path from Mq to M, For example, the path MomdefM may be reduced to a combination of the paths M^^mdMo) ModeMo, MoefMoy and M^fM. The path M^mdM^ may then be reduced to a loop-circuit about the singular point a, and similarly for the other two. Finally, the path M^fM is equivalent to a direct path. It follows that, whatever be the path of integration, the value of the line integral will be of the form (52) F(x, y) = Fix, y) -f mA -f tiB + ^C, where m, n, and p may be any positive or negative integers. The quantities A, B, C are called the periods of the line integral. That integral is evidently a function of the variables x and y which admits of an infinite number of different determinations, and the origin of this indetermination is apparent. Remark, The function F{x, y) is a definitely defined function in the whole region A when the cuts aa, bfi, cy have been traced. But it should be noticed that the difference F(m) - F(rriJ) between the values of the function at two points m and m' which lie on opposite sides of a cut does not necessarily vanish. For we have ' + / + I , M^ Jm Jmf which may be written But/Ji«zero; hence F(^-F(^= A. It follows that the difference F{rn) - F{m^ is constant and equal to A all along aa. The analogous proposition holds for each of the cuU. VII, JIM] TOTAL DUFERENTIALS «fl Example. T^^** Ti»'« integral 4/(1.1 xdy — ydx '0.0, x' + y* has a single critical point, the origin. In ortler to find tlw spending period, let us integrate along the circle jt'-I-^m^. Along this circle we have x = pcosw, y = psin«u, xdy - ydx = p'^dm, whence the period is equal to ^*V« = 2ir. It is easy to Terify this, for \hiy intf^trrand is the total differential of arotmny/z. 154. Common roots of two eqaations. Let X and T be two fanctloiMof llw variables z and y which, together with their first partial deriTatlves, a tinuous in a region A bounded by a single eloeed contour C. Then the 8ion {XdY - YdX)/{X^ + Y*) satisfiee the oondiUon of integrabUitj, for U to the derivative of arc tan Y/Z. Hence the line Integral (53) , ZdT-TdX I taken along the contour C in the poeitive sense vaotohea cients of dx and dy in the integrand remain continoous Insida C, La if ths two curves X = 0, y = have no common point inside that oootomr. But tf tlMst two curves have a certain number of common points a, ft, c, • • • TnsMt C, of the integral will be equal to the sum of the values of ths i along the circumferences of small circles described about tbs potots a, A^ e, • • • m centers. Let (a, /3) be the coordinates of one of the comiiKNl poiala Wa skall suppose that the functional determinant D(X, Y)/D{z, y) to not asro. La. llMi the two curves X = and F = are not Ungent at the point. Tbsa it to poa-> sible to draw about the point (a, fi) as center a circle c whose radios to so saaU that the point (X, Y) describes a small plane region about ths point (0, 0| which is bounded by a contour y and whicb oo r rssponds poiai for poteS to tiM circle c (§§ 25 and 127). As the point (x, y) describes the circumfSrenos of tka olroto e la ths sense, the point (X, Y) describes the contour > In the positive or la tbs sense, according as the sign of the functional datanaiBaDt iarida tbs etaela s to positive or negative. But the deflnita Integral sloag the drcnadmrnmoi e to equal to the change in arc tan Y/X in one revolution, that to, ± reasoning for all of the roots shows that / <"> L^^^="<^-'>- where P denotes the number of poinds common to ths two eorres at whfcb D(A', Y)/D{x, y) is posiUve, and .Y ths number of ooonaon polato al whteb tbs determinant is negative. 822 MULTIPLE INTEGRALS [VII, §155 Hm definite integral on the left is also equal to the variation in arc tan Y/X fat going aruuud c, that is, to the index of the function YJX as the point (x, y) dMoribet the contour C If the functions X and Y are polynomials, and if the oontoor C is composed of a finite number of arcs of unicursal curves, we are led to calculate the index of one or more rational functions, which involves only elementary operations (§ 77). Moreover, whatever be the functions X and Y^ we can always evaluate the definite integral (54) approximately, with an error lees than r, which is all that is necessary, since the right-hand side is always a maltiple of In. The fonnula (64) does not give the exact number of points common to the two cur>'e« unless the functional determinant has a constant sign inside of C Picard*8 recent work has completed the results of this investigation.* 199. Gmieralixation of the preceding. The results of the preceding paragraphs may be extended without essential alteration to line integrals in space. Let P, Q, and /? be three functions which, together with their first partial derivatives, are continuous in a region (E) of space bounded by a single closed surface iS. Let OS seek first to determine the conditions under which the line integral (66) Cr= / P(to + Qd?/ + -Bd2 depends only upon the extremities (xo, yo, z^ and (x, y, z) of the path of inte- gration. This amounts to inquiring under what conditions the same integral ranishes when taken along any closed path r. But by Stokes' theorem (§136) the above line integral is equal to the surface integral //(S-5)"-(s-'>-(f-S>' extended over a surface S which is bounded by the contour r. In order that this surface integral should be zero, it is evidently necessary and sufficient that the equations (6«) sP_aQ aQ_aB ^_5^ ay ax * Zz~ dy' dx ~ dz ihoald be satisfied. If these conditions are satisfied, U" is a function of the vari- ables X, y, and z whose total differential is P dx + Q dy + iJ dz, and which is single valued in the region {E). In order to find the value of U at any point, the path of integration may be chosen arbitrarily. If the functions P, Q,and R satisfy the equations (66), but at least one of them becomes Infinite at all the pomts of one or more curves in (JK), results analogous to those of § 153 may be derived. If, for example, one of the functions P, Q, R becomes infinite at all the points of a closed curve 7, the integral U will admit a period equal to the value of the line integral Uken along a closed contour which pierces once and only once a •orfaoe r bounded by 7. We may also consider questions relating to surface integrals which are exactly analogous to the questions proposed above for line integrals. Let ^, B, and C be three functions which, together with their first partial derivatives, are •TraU4 d'AnatyBe, \o\.l\. Vll.fWfi] TOTAL D1FFERKNTIAL8 contlnuouA in a region(JSr)of >p«oeboaiidMlb7adjifUeloMdMrteM& LkZ be a surface inside of (E) bounded by a oootour r of any form whaieTtr. the surface integral (67) ^=rr Ad^dg-^Bdxdx-i-Cdzdv J J(2) depends in general upon the surface Z as well as upon tha eonloar r. la that the integral should depend only upon r, it la eridaDtly naosHary aad cient that its value when taken over any closed stufiM In (JT) sliottld vaaWi. Green's theorem (§ 140) gives at once the conditions under wlUeh this Is tma. For we know that the given double integral extended orer any clowd nrfaea ii equal to the triple integral ///(g*5'*f) dx<^dg extended throughout the region bounded by the wirffww In ofder lluii tiili litlar integral should vanish for any region inside (i?), it is evidently necessary that Um functions Ay B, and C should satisfy the equation This condition is also sufBcient. Stokes' theorem affords an easy verlficatioD of this fact. For If A^ B, and C are three functions which satisfy the equation (68), It is always possible to deter- mine in an infinite number of ways three other functions P, Q, and R (69) ««_£« = ^. »l-»Ji^B. ?«-»f=C. ^ ' By dz H dz to «y In the first place, if these equations admit solutions, they admit an number, for they remain unchanged if P, Q, and R be replaeed by tx ty H respectively, where X Is an arbitrary function of x, y, and t. Again, R=Q, the first two of equations (60) gire P= C B(x, y, z) dz + 0(x, y) , Q = - f 'yl(x, y, «)ds + f (a. f). where <p{x, y) and v('(x, y) are arbitrary functions of z and y. values in the last of equations (50), we find a-g.cB.,.«, or, making use of (58) One of the functions or ^ may still be chosen at The functions P, Q, and R having been deCermioed, the sortMa latsfral, kj Stokes' theorem, is equal to the Une integral /tn^^*^ Q^ + **• •**^'*^ evidently depends only upon the oontoor V. SSi MULTIPLE INTEGRALS [VII, Exs. EXERCISES 1. Find the valae of the triple integral r r r[6(x - yy + Sae - ia^]dxdydz Vt^ m^'^ throughout the region of space defined by the inequalities x« + y«-a«<0, xa + y2_|.22_ 2a2<0. [Licence^ Montpellier, 1896.] 2. Find the area of the surface ^ ^ ^ a2x2 + 62y2 and the yolume of the solid bounded by the same surface. 3. Inrestigate the properties of the function F{X, Y, Z) = f'dx ( dy( /(x, y, z)dz eootidered as a function of X, F, and Z. Generalize the results of § 125. 4. Find the volume of the portion of the solid bounded by the surface (x2 + y2 + z2)8 _ Z(j%xyz which lies in the first octant. 6. Reduce to a simple integral the multiple integral //' * '/*''^»' ' ■ ' ^«"^(^i + X2 + • . . + x„)(toi(te2. . -dxn extended throughout the domain D defined by the inequalities O^xi, 0<x,, •.., 0<x„, xi + X2 + --.+x„<a. [Proceed m in § 148.] 6. Reduce to a simple integral the multiple integral extended throughout the domain D defined by the inequalities OS... o^x.. .... o<x., (5!y' + ... + (5.y"<i. ?•. DeriTe the formula M JJJ J r(|+i) VII. Em.] exercises where the multiple integral It eztMuitxi Uirougooui Um dooMln Ddttead ky Ite Inequality 8*. Derive the formula f (10 f F{a COB $ -k- b ain$ oo§^ -^ e tin $ Bin ^) tin 0d^stwf^^F{uil) dm, where a, 6, and c are three arbitrary oonaUnu, and where B = VSHTPT?. [PotaM*.] [First observe that the given double integral la eqoal to a oartaia aorteee IbI»> gral taken over the surface of the sphere 2* + y* + S* s 1. Thai take tba plaaa tu; + ^2/ + cz = as the plane of zy in anew ayiteni of coOntlaaUa.] 9*. Let p = F(d, <f>) be the equation in polar oo<}nllnatee of a Show that the volume of thn solid bounded by the eurfaoe ia equal to tte itrrahli integral {a) IjfpCOByda extended over the whole surface, where da repreeenti the element of area, and 7 the angle which the radius vector makes with the exterior normaL 10*. Let us consider an ellipsoid whoae equation it — 4- ' 4. = 1 and let us define the positions of any pohit on its sorfaoe by the aOipCle oo(MI- nates v and p, that is, by the roots which the aboTO equation wonld lMf« K ^ were regarded as unknown (cf. § 147). The application of the liwnia (tt) •• the volume of this ellipsoid leads to the equation Likewise, the formula (a) gives Jq Jb V(6*-/^(ci-/^(»«-M)(«^-i^ • [La«i.) 11. Determine the funcUons P(x, y) and Q(x, y) which, togeClMr wlUi ihaU partial derivatives, are continuous, and for which the line iniegrai JPC* + a, y + « d« + Q(« + a, y + A) % token along any closed contour whateTer ia tnJa p w i d wi t of Um ooMtanta a and B and depends only upon the contour itself. ^ [Lk«c«, Pafta. July. 1900.] 826 MULTIPLE INTEGRALS [VII, Exs. 18^. Consider the point transformation defined by the equations y = 0(x', y', 2') , s = f(x', 1/^,2'). As the point (a^, y*, zO describes a surface S% the point (x, j/, z) describes a sur- face 5. Let a, ^, 7 be the direction angles of the normal to S ; a\ p\ Y the direction angles of the corresponding normal to the surface 8' ; and da and da' the corresponding surface elements of the two surfaces. Prove the formula COS7d.=±d<r-{ j^^>^ cosa- + ^<^cos/3- + ^<^cosy}. 13*. Derive the formula (16) on page 304 directly. [The volume V may be expressed by the surface integral V= f z cosy da, and we may then make use of the identity D{x', y'^zf) dx^r D(y\ zf)) dir D{z\ X') ) dz'V B{x\ y') j which is easily verified.] CHAPTER VIII INFINITE SERIES I. SERIES OF REAL CONSTANT TERMS GENERAL PROrERTIKS TESTS FOR CONVERGENCE 156. Definitions and general principles. Seqaenoet. The elementarj properties of series aie discussed in all texU on College Algebra and on Elementary Calculus. We shall review rapidly the principal points of these elementary discussions. First of all) let us consider an infinite tequence uf qumn titles (1) «o, «i, «s in which each quantity has a definit< rder of preoedeoea hQm^ fixed. Such a sequence is said u, .. , -j^fU if #, approftehet a limit as the index n becomes infinite. Every sequence which is not convergent is said to be divergent. This may happen in either of two ways : s^ may finally become and remain larger than any preassigned quantity, or s^ may approach no limit even though it does not become infinite. In order that a sequence should be convergent, it is sufficient thaty corresponding to any preoMigned poeUive nwmker c, a positive integer n should exist such that the difference «,♦» — ^ *• less than c in absolute value for any positive integer p. In the first place, the condition is necessary. For if j^ approacfaat a limit s as n becomes infinite, a number it always exists for whieh each of the differences < — »,, * — *■>!» ••» *—**♦#> •••is lass than </2 in absolute value. It follows that the absolute value of #, ^ , will be less than 2 c/2 = c for any value of p. In order to prove the converse, we shall introdnoe a v«ry impor> tant idea due to Cauchy. Suppose that the absolute value of eaeh of the terms of the sequence (1) is less than a positive number AT. Then all the numbers between — N and + N may be separated into two classes as follows. We shall say that a number balangi lo Um class .4 if there • vJ^» an infinite number of terms of the sequtnoe (1) :W7 828 INFINITE SERIES [VIII, §166 which are greater than the given number. A number belongs to the claai B if there are only a finite number of terms of the sequence (1) which are greater than the given number. It is evident that every number between - N amd -hN belongs to one of the two classes, and that every number of the class A is less than any number of the class B. Let S be the upper limit of the numbers of the class A, which is obviously the same as the lower limit of the numbers of the class B. Cauchy called this number the greatest limit {la plus grande des limites) of the terms of the sequence (1).* This number S should be carefully distinguished from the upper limit of the terms of the sequence (1) (§ 68). For instance, for the sequence .111 ^' 2' 3^ •"' n '" the upper limit of the terms of the sequence is 1, while the greatest limit is 0. The name given by Cauchy is readily justified. There always exist an infinite number of terms of the sequence (1) which lie between S — € and .S -|- c, however small c be chosen. Let us then consider a decreasing sequence of positive numbers ci, cg, •••, c,, •••, where the general term «„ approaches zero. To each num- ber €i of the sequence let us assign a number a,, of the sequence (1) which lies between -S — €,• and S + Ci. We shall thus obtain a suite of numbers ai, a^, •••, a„, ••• belonging to the sequence (1) which approach S as their limit. On the other hand, it is clear from the very definition of S that no partial sequence of the kind just mentioned can be picked out of the sequence (1) which approaches a limit greater than S. Whenever the sequence is convergent its limit is evidently the number S itself. Let us now suppose that the difference s„^p — s„ of two terms of the sequence (1) can be made smaller than any positive number c for any value of /> by a proper choice of n. Then all the terms of the sequence past s^ lie between s„ — e and s„ -\- c. Let S be the greatest limit of the terms of the sequence. By the reasoning just given it is possible to pick a partial sequence out of the sequence (1) which approaches S as its limit. Since each term of the partial •aqaeoce, after a certain out, lies between s^ — c and s„ -\- «, it is •n4mtm4» analyHques de TuHn, 19SS (CoUetted Works, 2d series, Vol. X, p. 49). The daflnltloD may be extended to any aasemblage of numbers which has an upper nsilt. VI11,§1S7J CONSTANT TKKIIg S£9 clear that the absolute value of ^ — «. is at mott eqita] to c Nov let «» be any term of the sequence (1) whoee index m is pnotm than n. Then we may write and the value of the right-hand side is surely less than 2c Sinea t is an arbitrarily preassigned positive number, it follows that the general term «. approaches 5 as its limit as the index m increases indefinitely. Note. If S is the greatest limit of the terms of the sequence (1), every number greater than S belongs to the class B, and OTery num- ber less than S belongs to the class A. The number S itnlf may belong to either class. 157. Passage from sequences to series. Given any infinite saquenea Wo, Wj, 1/„ .., tt,, -.-, the series formed from the terms of this sequence, (2) «o + «i + Wt + - • + «. + •••, is said to be convergent if the sequence of the saocaaaiTa stuns So = yo, Si = UQ + ni, ..., 5, = tto-f-tti + --- + ti„ ••• is convergent. Let S be the limit of the latter sequence, ie. the limit which the sum S^ approaches as n increases indefinitely: trmm 5 == lim 5, = lim V «». "-• "-•^ Then S is called the sum of the preceding serieSf and this retatioii is indicated by writing the symbolic equation 5 = Wo4-Wi +• -f w. -f •••=X"''* A series which is not convergent is saia to oc diver^emL It is evident that the problem of determining whalher the aariaa is convergent or divergent is equivalent to the problem of determin- ing whether the 8e<iuenoe of the successive sums ^, Si, ^, ••• ia convergent or divergent. Conversely, the sequenoa *0, 'l» *•> "'» *•' will be convergent or divergent aoeording as the sariaa 5o -H (»i -«•) + («t - «i) +•+(«•-*—») + ••• 380 INFINITE SERIES [VIII, §167 is convergent or divergent For the sum S^ of the first n + 1 terms of this series is precisely equal to the general term s^ of the given sequence. We shall apply this remark frequently. The series (2) converges or diverges with the series (3) w, + w,+i + --- + Wp+, + ---, obtained by omitting the first p terms of (2). For, if S^(n >^) denote the sum of the first » + 1 terms of the series (2), and 2„_p the sum of the n — /> + 1 first terms of the series (3), i.e. the difference 5, - 2,_^ = i^^ + wj + • • • + w^.i is independent of n. Hence the sum \_p approaches a limit if 5„ approaches a limit, and conversely. It follows that in determining whether the series converges or diverges we may neglect as many of the terms at the beginning of a series as we wish. Let 5 be the sum of a convergent series, S^ the sum of the first n + 1 terms, and R^ the sum of the series obtained by omitting the first n -f 1 terms, K = ^^ + 1 + Un + 2 H f ^n + p H • It is evident that we shall always have 5 = 5„ + R,, It is not possible, in general, to find the sum 5 of a convergent series. If we take the sum S of the first n -{-1 terms as an approxi- mate value of S, the error made is equal to R„. Since S„ approaches 5 as n becomes infinite, the error R^ approaches zero, and hence the number of terms may always be taken so large — at least theoret- ically — that the error made in replacing S by 5„ is less than any preassigned number. In order to have an idea of the degree of approximation obtained, it is sufficient to know an upper limit of /?,. It is evident that the only series which lend themselves readily to numerical calculation in practice are those for which the remainder R^ approaches zero rather rapidly. A number of properties result directly from the definition of con- rergenoe. We shall content ourselves with stating a few of them. 1) y each of the terms of a given series be multiplied by a constant k different from zerOf the new series obtained will converge or diverge with the given series; if the given series converges to a sum S, the sum qfthe seeond series is kS, VIII. §138] CONSTANT TERlfg |g| 2) If there he given two eonvergent series «o + Mi + «« + •• + «, + ..., Vt4-Vi + », + - • + V. + --, whose sums are S and S*, respeetivelf, the new smist eUaim^d hp adding the given series term hy term, K -f t;^) + (th + v,)+ ••• +(u. + tf.)+ • converges, and its sum m 5 -f 5'. The analogous theorem hoUU for the term-by -tenn addition of p convergent series. 3) The convergence or divergence of a series is not affeeted if the values of a finite number of the terms he changed. For such a change would merely increase or decrease all of the sums S^ after a oertain one by a constant amount. 4) The test for convergence of any infinite sequence, applied to series, gives Cauchy^s general test for convergence : • In order that a series he convergent it is neeessarg and thatf corresponding to any preassigned positive number c, a n should exist, such that the sum of any number of terms what- ever, starting with w,^.i, is less than c in abeoiute vaiu$. For ^n + p - Sn = ^^» + I + W,+, H + W«+p. In particular, the general term w,^, = 5,^, — 5, must mppremek zero as n becomes infinite. Cauchy's test is absolutely general, but it is often difficult to apply it in practice. It is essentially a development of the ftrj notion of a- limit. We shall proceed to recall the practical rules most frequently used for testing series for convergenoe and dirergeiiee. None of these rules can be applied in all oases, bat together they suffice for the treatment of the majority of cases which actually arise. 158. Series of positive terms. We shall oommenoe fay investigatiiif a very important class of series. — those whose terms are all posi- tive. In such a series the sum 5. increases with n, Henee in order that the series converge it is sufficient thai the sum 5. should remain less than some fixed number for all values of m. The most general test for the convergence of such a series is based upon com- parisons of the given series with others prerionsly studied. Tb* following propositions are fundamental for this process: ^£x«rctoM de iral/UmoMYMif, 1817. (OoUmCmI ITtrte, VoL Til, M 8S2 INFINITE SERIES [vm,§159 1) If each of the terms of a given aeries of positive terms is less than or at most equal to the corresponding term of a known convergent weriet of positive termsy the given series is convergent For the sum 5, of the first n terms of the given series is evidently less than the sum S* of the second series. Hence S^ approaches a limit S which is less than S\ 2) If each of the terms of a given series of positive terms is greater than or equal to the corresponding term of a known divergent series of positive termSy the given series diverges. For the sum of the first n terms of the given s^ies is not less than the sum of the first n terms of the second series, and hence it increases indefinitely with n. We may compare two series also by means of the following lemma. Let (U) Mo + Wj + w, + ... + w„4...., (F) t;^-|-Vj+t;^ + ...+V^ _!_... be two series of positive terms. If the series (U) converges, and if, after a certain term, we always have v„+i/v„ ^ Wn+i/w„, the series ( V) also converges. If the series (U) diverges, and if, after a c&rtain term, we always have u^+i/u„<v^+i/v„, the series (F) also diverges. In order to prove the first statement, let us suppose that ^•-^i/^'K+i/^n whenever n>p. Since the convergence of a series is not affected by multiplying each term by the same con- stant, and since the ratio of two consecutive terms also remains unchanged, we may suppose that v^ < u^, and it is evident that we should have v^^,<u^^^, Vp^,<u^^„ etc. Hence the series (F) must converge. The proof of the second statement is similar. Given a series of positive terms which is known to converge or to diverge, we may make use of either set of propositions in order to determine in a given case whether a second series of positive terms converges or diverges. For we may compare the terms of the two series themselves, or we may compare the ratios of two eonsecutiye terms. 159. Ctuchy's test and d'Alembert'B test. The simplest series which OMi be used for purposes of comparison is a geometrical progression whose ratio is r. It converges if r < 1, and diverges if r > 1. The eomptrison of a given series of positive terms with a geometrical progression leads to the following test, which is due to Cauchy: VIII, 5159] CON'STAN'T TERMR $|g If the nth root "wu^ qf the general term u^ qf a smrise ^ terms after a certain term is eonatanily U$s tJkmi a fiaitd than unity, the series converges. If y/u^ after a Mvioiw term is cmi- stantly greater than unity, the series diverges. For in the first case 'Vu,<k<l, whence M.<Jb". Heooe eeeh of the terms of the series after a certain one ie leti than the oon^ sponding term of a certain geometrical progreeeion whoee ratio ie less than unity. In the second case, on the other hand, \^>1, whence u^>l. Hence in this caae the general term doee not approach zero. This test is applicable whenerer "V^ i^proachea a limit Id fact, the following proposition may be stated: If v^ approaches a limit I as n heeomes infinite, ths ssriss wiU converge if I is less than unity j and it will diverge \flis greater tAsm unity. A doubt remains if lalj except when "Vu^ remains greaUr ikms unity as it approaches unity, in which mm the series surely div erfs t. Comparing the ratio of two consecatiye terms of a given series of positive terms with the ratio of two consecutiTe termi of a geometrical progression, we obtain d'Alembert's test: If in a given series of positive terms the ratio of any term te tMe preceding after a certain term remains less than a fixed less than unity, the series converges. If thai raiie softer • term remains greater than unity, the series div erg es * From this theorem we may deduce the following corollary: If the ratio u^^x/u^ approaches a limit I as n heoomee infinite, the series converges if I <!, and diverges tfl>l. The only doubtful case is that in which / « 1 ; a»eii then, i^t^4.i/*^ remains greater than unity as it approaehes umiig, the series is diesrge^ General commentary. Cauchy*s fast is BOffe gMMnl tbaa d*AlMBbert*t. Wee suppose that the terms of a given miM, iflar a osftain one, art seek Its Ikea the corresponding terms of a decreasliig gsooMliieal iNUgiMrinn. Le. that Ike general term u« is leas than Ar* for all TihiM of a grealertheaa ftsed liilli»F> where A is a certain constant and r is lea than nnhy. Henoe V^ < MW*, aad the second member of thia inequality a|>proeohM uatty •• m bMomta leMeSm. Hence, denoting by fc a fixed number between r and 1, we shall have after a eer- tain term Vu^<k. Hence Cauchy's test Ie appUeable In any aneh eeee. Bal ll may happen that the raUo u, + i/m, assumes ralnee fraater than unity. far out in the series we may go. For example, consider the l + r|alna| + r«|ain2a| + .-- + »«|sln«a| + 3g4 INFINITE SERIES [VIII, §169 where r < 1 and where a ia an arbitrary conatant. In this case V^ = r V|sinna| < r, whereat the ratio Un±i_^ 8in(n + l) a tin sin na may aKome, in general, an infinite number of values greater than unity as n iocreaeee indefinitely. Nererthelen, it is advantageous to retain d'Alembert's test, for it is more convenient in many cases. For instance, for the series 1 1.2 1.2.3 1.2-. n the wUo of any term to the preceding is x/{n +1), which approaches zero as n becHwneff infinite ; whereas some consideration is necessary to determine inde- pendently what happens to Vun = x/\/l . 2 • • • n as n becomes infinite. After we have shown by the application of one of the preceding tests that each of the terms of a given series is less than the corresponding term of a decreasing geometrical progression A, Ar^ Ar^, • • • , ^r», • • • , it is easy to find an upper limit of the error made when the sum of the first m terms is taken in place of the sum of the series. For this error is certainly less than the sum of the geometrical progression Ar^ Ar^ + Ar^+^ + A7^+^ + '-' = 1 — r When each of the two expressions -n/w^ and Vn + i/Un approaches a limit, the two limits are necessarily the same. For, let us consider the auxiliary series (4) Uo-\-UiX-\- UiZ^ + • • • + lAnX** + • • • , where x is positive. In this series the ratio of any term to the preceding approaches the limit Ix, where I is the limit of the ratio w„ + i/m„. Hence the ■eriee (4) converges when x < 1/i, and diverges when x > l/l. Denoting the limit of v^ by T, the expression y/unX'* also approaches a limit Vx, and the series (4) converges if x < l/l\ and diverges if x > 1/V. In order that the two tests should not give contradictory results, it is evidently necessary that I and /' should be equal. If, for instance, I were greater than T, the series (4) would be convergent, by Cauchy's test, for any number x between l/l and 1/V, whereas the aame series, for the same value of x, would be divergent by d'Alembert's test. Still more generally, if tt, + i/u„ approaches a limit Z, v^ approaches the same limit.* For suppose that, after a certain term, each of the ratios ^ + 1 Un + t _ Un + p llee between I - c and I + e, where e is a positive number which may be taken M tmaU as we pleaee by uking n sufficiently large. Then we shall have u:*''(i-.)-+p< 'yy^^^ < u:+p(i + .)-+i>. •Cauchy, Court d* Analyse, VIII, f 160] CONSTANT TSRMS t|6 As the number p inoretaM indefinitely, while » remaiiMi ftsed, the two leran oa the extreme right and left of Uiie double inequaUty appriMeh I -f • Mid I ~ c, respectively. Hence for all raluee of m greater than a nilablj we ihall have and, since e is an arbitrarily assigned number, it follows that K(m^ tlie number / as it« limit. It should be noted that the converse is not true. Oonalder, for sequence 1, a, 06, cflby a«6«, •., o"ft^-», ••6>», , where a and b are two different numbers. The ratio of any term to the praeed- ing is alternately a and 6, whereas the expression \(u^ approaehss the Uailt V3 as n becomes infinite. The preceding proposition may be employed to determine the UmUs of Mr- tain expressions which oc cur in undetermined forms. Thos it Is srldeBt that the expression v^l .i-n increases indefinitely with a, since the ratio a !/(a - 1)1 increases indefinitely with n. In a similar manner it may be shown that each of the expressions -v^ and \^logn approachee the limit unity as a 160. Application of the greatest limit. Cauchy formolated the in a more general manner. Let eu be the general term of a terms. Consider the sequence 1 1 ! (5) au <4, fl{, •••, K* •••• If the terms of this sequence have no upper limit, the gSMral approach zero, and the given series will be divergenL If aU the tenas of the sequence (5) are less than a fixed number, let w be the gi es t es l limit of the larms of the sequence. The series Za^ is convergent iftiUUta ikom wiif|f, ami M wmrgm t ifttiM than unity. In order to prove the first part of the theorem, let 1 - a be a aomber w and 1. Then, by the definition of the greatest limit, there exist bat a number of terms of the sequence (6) which are greater than 1 - a. It follows that a positive integer p may be found such that •Voi < 1 - a for all values of a greater than p. Hence the series So. converges. On the other hand. If •» > I, let 1 + a be a number between 1 and «. Then there are an Infiaita aambsr ef terms of the sequence (6) which are greater than I + a, and heooe there are aa infinite number of values of n for which o. is greater than nnl^. U foOofis that the series Sa, is divergent in this case. Tbeeass in which •• « 1 rsmatos ladowht. 161. Cauchy 's theorem. lu case Hi^i/i*. wd Vu^ both approMh unity without remaining constantly gnater than unity, neither d'Alembert's test nor Cauchy's test enables tie to decide whether the series is convergent or divergent We moft then take M a comparison series some series which has the sane oharaMerMo 886 INFINITE SERIES [VIII, § 161 but which is known to be convergent or divergent. The following proposition, which Cauchy discovered in studying definite integrals, often enables us to decide whether a given series is convergent or divergent when the preceding rules fail. Let ^(j-) be a function which is positive for values of x greater than a certain number a, and which constantly decreases as x increases past x — a, approaching zero as x increases indefinitely. Then the x axis is an asymptote to the curve y = </>(x), and the definite integral jT <f>(x)dx may or may not approach a fiinite limit as I increases indefinitely. The series (6) i>(a) 4- ,^(a + 1) + • • • + <^(a + w) + • . . converges if the preceding integral approaches a limit, and diverges if it does not. For, let us consider the set of rectangles whose bases are each unity and whose altitudes are <f>{a), <f>{a +1), • • ., <^(a + n), respec- tively. Since each of these rectangles extends beyond the curve y = ^(x), the sum of their areas is evidently greater than the area between the x axis, the curve y = <i>{x), and the two ordinates x = a, jc = a -f n, that is, <t>(x)dx. On the other hand, if we consider the rectangles constructed inside the curve, with a common base equal to unity and with the altitudes «^(a 4-1), <t>{a -f 2), • • ., <^(a + n), respectively, the sum of the areas of these rectangles is evidently less than the area under the curve, and we may write ^(a) 4- ^(a +1) -f • • • + ^(a 4- n) < <^(a) 4- f" ^<f>(x)dx. Hence, if the integral X'<^(a;) rfa; approaches a limit L as I increases indefinitely, the sum ^(a) + ... +^(a-{-n) always remains less than ^(a) 4- L, It follows that the sum in question approaches a limit ; hence t^ series (6) is convergent. On the other hand, if the inte- ff^f^^*H^)dx increases beyond all limit as n increases indefinitely, the same is true of the sum *(«) 4- ^(a 4- 1) 4- • • • -f <^(a 4- n), VIII, § 161] CONSTANT TERMS 817 as is seen from the first of the above inequAlitiat. Hene^ in thif case the series (G) diverges. Let us consider, for example, the function 4(x)v]/j^, whn% ft is positive and a =1. This function satisfies all the requiienms of the theorem, and the integral j^\\/af^'\dx appxoacbee a limit •• / increases indefinitely if and only if ft is greater than unity. It follows that the series p + 2^ + 3;r + --- + jjj + --- converges if /i is greater than unity, and diverges if ft <1. Again, consider the function ^(2)sl/[x(logx)''], where log* denotes the natural logarithm, /a is a positive number, and awm%. Then, if /* =^ 1, we shall have X' dx —1 ___ = __[(log„).-._(,og2).-]. The second member approaches a limit if ft > 1^ indefinitely with n if /a < 1. In the particular case when /kmI it is easy to show in a similar manner that the integral beyond all limit. Hence the series 2(log 2Y ^ 3(log 3r ^ ^ n(log n^ ^ converges if /i > 1, and diverges if fi<l. More generally the series whose general term is 1 n log n log« n log* n • • 1(^'~* »(log' nf converges if /i > 1, and diverges if /* ^ 1. In this expression log*» denotes log log n, log' rt denotes log log log n, etc. It is understood, of course, that the integer n is given only valoes so large that logn, log^w, log«n, ••, log^n are positive. The missing terat in the series considered are then to be supplied by aeros. The theorem may be proved easily in a manner similar to the demoo- strations given above. If, for instance, fi¥»l, the funotioo 1 X log a; log* X. •(log' 'V* is the derivative of (log" x)> -**/(! - m)» ^^ ^^ *»«^^ tuncttoo approaches a finite limit if and only if ^ > 1. 888 INFINITE SERIES [VIII, §162 Caocby*! theorem admits of applications of another sort. Let us suppose that the function ^(z) satisfies the conditions imposed above, and let us con- ^(n) + 0(n + 1) + . . . + ^(n + p), where ii and p are two integers which are to be allowed to become infinite. If the whoM general term is 0(n) is convergent, the preceding sum approaches •• a limit, since it is the difference between the two sums Sn + p+i and Sn, of which i^proaches the sum of the series. But if this series is divergent, no conclusion can be drawn. Returning to the geometrical interpretation given ahove, we find the double inequality JT" %{x)dx < 0(n) + 0(n + 1) 4- • • • + 0(n + p) < <f>{ny+ f'"^%{z)dx. Since 0(n) approaches zero as n becomes infinite, it is evident that the limit of the sum In question is the same as that of the definite integral f*^'^^<p{x)dx, and this depends upon the manner in which n and p become infinite. For example, the limit of the sum \ 1 +...+ » n n + 1 n + p to the nme ae that of the definite integral f^'^^ [1/x] dx = log(l + p/n). It is etear that this integral approaches a limit if and only if the ratio p/n approaches a limit. If a is the limit of this ratio, the preceding sum approaches log (1 + a) M Ite limit, as we have already seen in § 49. Finally, the limit of the sum Vn Vn + l Vn+p ft the aame as that of the definite integral s: la order that this expression should approach a limit, it is necessary that the ratio p/Vn should approach a limit or. Then the preceding expression may be written hi the form = 2 P yTn and It la erident that the limit of this expression is a. 192, Logarithmic criteria. Taking the series M a omnparison series, Caucliy deduced a new test, for convergence wWoh U entirely analogous to that which involves </u^. VIII, §162] CONSTANT TKRMS ggg If after a certain term tht- tj-jir-' \ ',)/logJi if alwayg greater than a fixed number which t^ j,''". r i',<in unity ^ ike serts» converges. If after a certain term log(l/Mj/logn w aUoojfg Ut$ than unity y the series diverges. If log(l/tt^)/logn approa^shes a limit t <u n increases ind^nMf, the series converges if / > 1, and diverges if / < 1, The ease m which / = 1 remains in doubt. In order to prove the first part of the theorem, we will remaik that the inequality log— > k log n is equivalent to the inequality — > n* or w_ < -r ; "since A; > 1, the series surely converges. Likewise, if log— < log n, we shall have ?/„ > I/71, whence the series surely diverges. This test enables us to determine whether a given series con- verges or diverges whenever the terms of the series, after a certain one, are each less, respectively, than the corresponding terms of the series wh ere A is a constant factor and /a > 1. For, if ''-^^' we shall have 5 log u^ 4- /x log n < log A or logn ^ ^ logn' and the right-hand side approaches the limit /ui as n im indefinitely. If A' denotes a number between unity and ^ we shall have, after a certain term, rr—^>K. logn 340 INFINITE SERIES [VIII,§163 Similarly, taking the series Z^ n(log nY ' Z^ n log w(log'* w)" ' as comparison series, we obtain an infinite suite of tests for con- vergence which may be obtained mechanically from the preceding by replacing the expression log(l/w„)/log7i by log [l/(7iw„)]/ log^ ;i, then by , 1 log \ nu^ log n i log'w and so forth, in the statement of the preceding tests.* These tests apply in more and more general cases. Indeed, it is easy to show that if the convergence or divergence of a series can be established by means of any one of them, the same will be true of any of those which follow. It may happen that no matter how far we proceed with these trial tests, no one of them will enable us to determine whether the series converges or diverges. Du Bois-Reymond f and Pringsheim X have in fact actually given examples of both convergent and divergent series for which none of these logarithmic tests deter- mines whether the series converge or diverge. This result is of great theoretical importance, but convergent series of this type evidently converge very slowly, and it scarcely appears possible that they should ever have any practical application whatever in problems which involve numerical calculation. § 163. Raabe's or Duhamel's test. Retaining the same comparison series, but comparing the ratios of two consecutive terms instead of comparing the terms themselves, we are led to new tests which are, to be sure, less general than the preceding, but which are often easier to apply in practice. For example, consider the series of positive terms (7) Wo + «^ + wj H h w« H , • B«6 Bertnuid, Traiti de Calctd diffirerUiel et integral, Vol. I, p. 238; Journal dt lAouviUe, Ut series, Vol. VII, p. 35. t Ueber Cotwergmx von ReUien . . . (Crelle'a Journal, Vol. LXXVI, p. 85, 1873). I Attgnnelnt Tfuorie der Divergenz . . . (Sfathematische Annalen, Vol. XXXV, 1800). f In an example of a certain convergent series due to du Bois-Reymond it would bs nuc essa r y, according to the author, to take a number of terms equal to the volume <^f Ihe §arth eiqnreMted in cubic millimetert in order to obtain merely half the sum of Vlli,§ia3j CONSTANT TERMS S41 in which the ratio u.^i/u, approaches unity, remaining coiuuuiuj less than unity. Then we may write where a. approaches zero as n becomes infinite. The oomparisoD of this ratio with [n/{n -hi)]** leads to the following rule, discovered first by Raabe* and then by Duhamel.f If after a certain term the product na^ u alwayt grtaUr iktm m fixed number which is greaier than unity, the serisi eamver^. If after a certain term the same product is always Uu than uniiyf tk$ series diverges. The second part of the theorem follows immediately. For, since na^ < 1 after a certain term, it follows that 1 n and the ratio u^ + i/u^ is greater than the ratio of two conteeutiTe terms of the harmonic series. Hence the series diverges. In order to prove the Urst part, let us suppose that after a certain term we always have na^> k>l. Let /x be a number which lies between 1 and k, l</l<^^ Then the series surely converges if after a certain term the ratio u.^i/u. is less than the ratio [n/(n -1-1)]'' of two consecutive terms of the series whoee general term is n'*^. The necessary condition that this should be true is that (8) • ' - ' l+«. M' or, developing (1 -f l/n)** by Taylor's theorem limited to the term in l/7tS where A^ always remains less than a fixed number as n becomes infinite. Simplifying this inequality, we may write it in the fom • ZsUsohri/i /Br MaihmnatUt uttd Pk^ttk, Vol. Z, i Journal ds LknwOls, Vol. IV, 1888. 842 INFINITE SERIES [VIU, §163 The left-hand side of this inequality approaches /a as its limit as n becomes infinite. Hence, after a sufficiently large value of n, the left-hand side will be less than na^^ which proves the inequality (8). It follows that the series is convergent. If the product na^ approaches a limit ^ as w becomes infinite, we may apply the preceding rule. The series is convergent if Z>1, and divergent if /<1. A doubt exists if 1=1, except when na^ approaches unity remaining constantly less than unity : in that case the series diverges. If the product nan approaches unity as its limit, we may compare the ratio u«4.i/u» with the ratio of two consecutive terms of the series 1.1. 2 (log 2)'* n(logw)'^ which converges if /i>l, and diverges if ti<\. The ratio of two consecutive terms of the given series may be written in the form i*»i + . 1 n n where /3„ approaches zero as n becomes infinite. If after a certain term the product /3« log n is always greater than a fixed number which is greater than unity y the aeries converges. If after a certain term the same product is always less than unity, the aeries diverges. In order to prove the first part of the theorem, let us suppose that /3„ log n > A: > 1. Let M be a number between 1 and k. Then the series will surely converge if after a certain term we have w„ n + 1 Llog(n + l)J * which may be written in the form n n \ n/ L logn J or, applying Taylor's theorem to the right-hand side, n n \ n/( logn "L logn J ) wiMre Xa always remains less than a fixed number as n becomes infinite. Simplifying this inequality, it becomes / i\ ^«(n4-l)ri<>gCl + i')f Ailogm>M(n + l)log(l+M+-l- iLlLIiiLL. \ n/ logn VIII, JlG.i] CONSTANT TERMS M^ The product (n + 1) log(l -f i/n) approaches uitiiy m n btooow iBflallA, lor U may be writt«u, by Taylor*! theorem, in the form (10) (n + l)log/l + i^ = 1 + 1(1 + ,). where « approaches zero. The right-hand aide of the above meqaalitj limnian approaches fiMlia limit, and the truth of the inequality is nlililrtnil for nM cieiitly large values of n, since the left-hand tide ia greater than It, wbieb it HhU greater than fi. The second part of the theorem may be proved by oompartng the rafk> Un + i/Un with the ratio of two consecutive terma of the aerlea whoM fMietal term is l/(nlogn). For the inequality u, n + 1 log(n + l)' which is to be proved, may be written in the form n \ n/L logn J n n \ n/L log i /S„logn<(n + l)log (-!)■ The right-hand side approaches unity through valuea which are greater than unity, as is seen from the equation (10). The truth of the ineqoali^ is tbera* fore established for sufficiently large valuea of n, for the Muhand ikla eaBBOl exceed unity. From the above proposition it may be shown, as a ooroUaiy, that If the piod- uct /3n log n approaches a limit I as n becomes infinite, the l e ri ea ouu f ei gea if I >1, and diverges if ^ < 1. The case in which / = 1 remains in doabt, unlea ^ loga is always less than unity. In that case the series sorely divergea. If /3nlogn approaches unity through values which are greater than untey, wa may write, in like manner, Uw-t-l __ 1 «. ,+ ' + ' + ^-* n A logn where y^ approaches zero as n becomes infinite. It would then be posribia to prove theorems exactly analogous to the above bj o on a hfcifi ng the pcodMI 7„ log* n, and so forth. ,. Corollary. If in a series of posiUve terms the ratio of any term to the pra- c^ej^ng can be written in the form where m is a positive number, r a constant, and J?, a quantity whoae aha value remains less than a fixed number as n inereaaaa IndeOiihaly, (At tmlm verges if r ia greaUr than unity, and din^rgm te all Uker 844 INFINITE SERIES [vm,§164 For If we irt u«+i_ 1 we ihall haye Hn u* + i 1 Wn 1+i n logn n n n+1 n ^" n^ ,logn = 1- n »! + ** and hence llm lur, = r. It follows that the series converges if r > 1, and diverges If r < 1. The only case which remains in doubt is that in which r = 1. In order to decide this case, let us set From this we find and the right-hand side approaches zero as n becomes infinite, no matter how small the number fx may be. Hence the series diverges. Suppose, for example, that Un + i/Un is a rational function of n which ap- proaches unity as n increases indefinitely: Un-n _ nP 4- aiTU^^ -f (H^P"^ H Un ~ TIP + binP-^+b»7H^'^-\ * Then, performing the division indicated and stopping with thfrtatm in 1/n*, we may write u„ + i 1 . ai-h <f>{n) = 1 H 1 — , Un n n^ where 4>{n) is a rational function of n which approaches a limit as n becomes infinite. By the preceding theorem, the necessary and sufficient condition that the teriea should converge is that 6i > ai + 1 . This theorem is due to Gauss, who proved it directly.* It was one of the first for convergence. 164. Abtolute convergence. We shall now proceed to atudy series whose tenns may be either positive or negative. If after a certain term all the terms have the same sign, the discussion reduoes to the previous caae. Hence we may restrict ourselves to series which contain an infinite number of positive terms and an infinite * (OotUeUd Works, Vol. Ill, p. 138.) Disquisitioneit generates circa seriern ii\finitam a. a Mil, S m] CONSTANT TERMS S46 number of negative terms. We shall prore first of all the fol- lowing fundamental theorem : Ani/ series whatever is eanvergeni if the §&ri§i farmed ^f the akee- lute values of the terms of the given series eetwmr^ee. Let (11) «o-HMi-h---f K,4- be a series of positive and negatiye terms, and lei (12) f/^+f/j + .-.+ r, : be the series of the absolute values of the terms of the given seneg, where U^ = | Wh |- If the series (12) converges, the series (11) like- wise converges. This is a consequence of the general theorem of § 157. For we have and the right-hand side may be made less Uuui any prea.s8igned num- ber by choosing n sufficiently large, for any subeequant choice of p. Hence the same is true for the left-hand side, and the aeries (11) surely converges. The theorem may also be proved as follows : Let us write and then consider the auxiliary series whose general term is n, -f- T,, (13) (u, 4- ^o) + (1^ H- f^i) + ••• + (h, + tr.)+ •••. Let S^y S'^j and S'^ denote the sums of the first n terms of the series (11), (12), and (13), respectively. Then we shall have The series (12) converges by hypothesis. Hence the series (IS) also converges, since none of its terms is negative and its general term cannot exceed 2U^. It follows that each of the sums ^^ and S'^, and hence also the sum 5., approaches a limit as % increases indefinitely. Hence the given series (11) converges. It is evident that the given series may be thought of as arising from the guhtrsA- tion of two convergent series of positive terms. Any series is said to be absolutely o o nv e rgmi if the series of the absolute values of its terms converges. In smek a series the order ^ the terms may be changed in any way whatever %eithoui eJUrimg the 846 INFINITE SERIES [VIII, §164 sum of the series. Let us first consider a convergent series of posi- tiye termSi (14) ao + «iH -l-a«H 7 whose sum is Sy and let (16) *o + ^+-+*n4--- be a series whose terms are the same as those of the first series arranged in a different order, i.e. each term of the series (14) is to be found somewhere in the series (15), and each term of the series (15) occurs in the series (14). Let S'^ be the sum of the first m terms of the series (16). Since all these terms occur in the series (14), it is evident that n may be chosen so large that the first m terms of the series (15) are to be found among the first n terms of the series (14). Hence we shall have which shows that the series (15) converges and that its sum S' does not exceed S. In a similar manner it is clear that S ^ S'. Hence S' = S. The same argument shows that if one of the above series (14) and (15) diverges, the other does also. The terms of a convergent series of positive terms may also be grouped together in any manner, that is^ we may form a series each of whose terms is equal to the sum of a certain number of terms of the given series without altering the sum of the series* Let us first suppose that consecutive terms are grouped together, and let (16) ^, + ^^+^, + ...4.^^ + ... be the new series obtained, where, for example, ^0 = ^0 + «! + ••• + s, ^1 = S + i + • • • + a,, Then the sum S'^ of the first m terms of the series (16) is equal to the sum Sy of the first N terms of the given series, where N > m. As m becomes infinite, N also becomes infinite, and hence S'^ also approaches the limit S. Combining the two preceding operations, it becomes clear that any conmergerU series of positive terms muy be replaced by another series each of whose terms is the sum of a certain number of terms of the given series taken in any order whatever, without altering the sum of •UiM ofton Mid that parentheaes tnay be inserted in a convergent series 0/ positive in any inamur whatwer without altenng the sum 0/ the «erie«. — Trans. VIII. §lfi.n] CONSTANT TFRMS 34J the series. It is only ueoessary that each term of the giTen mtIm should occur in one and in only one of the vroupM whii h form the terms of the second series. Any absolutely convergent series may be regarded as the differ- ence of two convergent series of positive terms ; henee the praoedtiif operations are permissible in any such series. It is evident that tn absolutely convergent series may be treated from the point uf view of numerical calculation as if it were a sum of a finite number of terms. 165. Conditionally convergent series. A series whose terms do not all have the same sign may be convergent without being absolutely eon- vergent. This fact is brought out clearly by the following theoiem on alternating series, which we shall merely state, ftffuming that it is already familiar to the student.* A series whose terms are alternately positive and negative convergee if the absolute value of each term is less than that of the preeedimgp and if in addition^ the absolute value of the terms of the series diminishes indefinitely as the number of terms increases indefinitely. For example, the series converges. We saw in § 49 that its sum is log 2. The series of the absolute values of the terms of this series is precisely the harmonic series, which diverges. A series which oociTerges but which does not converge absolutely is called a eonditienaUy eemeet' (jent series. The investigations of Cauchy, Lejeune-Dirichlet, and Kiemann have shown clearly the necessity of distinguishing between absolutely convergent series and conditionally convergent aeriet. For instance, in a conditionally convergent series it is no^ always allowable to change the order of the terms nor to group the terms together in parentheses in an arbitrary manner. These operatioos may alter the sum of such a series, or may change a ooDraifaat series into a divergent series, or vice versa For exampla. l«t ui again consider the convergent series 111 1 1 2^3 4^ ^2n+l 2ii + 3 • It is iM>int^ out in § 166 that this therein to % wgeOsl esse of Um tlMoraai pnm4 there. — Trans. 848 INFINITE SERIES [vm,§l66 whose Bum ia evidently equal to the limit of the expression as m becomes infinite. Let us write the terms of this series in another Older, putting two negative terms after each positive term, as follows : ^ 2 4^3 6 8^ ^2n+l 4n + 2 4w + 4^ It is easy to show from a consideration of the sums S^^, S^^^^, and •^ta^i *^** *^® ^®^ series converges. Its sum is the limit of the expression 'vi^ I 1-] .T!)\2^+1 4n + 2 4w + 4/ as m becomes infinite. From the identity _JL 1 1_ = 1 /_J_ _ 1 \ 2n4-l 4n-f2 4w + 4 2\27i-hl 2n-\-2/ it is evident that the sum of the second series is half the sum of the given series. In general, given a series which is convergent but not absolutely convergent, It is possible to arrange the terms in such a way that the new series converges toward any preassigned number A whatever. Let Sp denote the sum of the first p positive terms of the series, and Sg the sum of the absolute values of the first q negative terms, taken in such a way that the p positive terms and the q negative terms constitute the first p + q terms of the series. Then the sum of the first p + q terms is evidently Sp — Sg. As the two numbers p and q increase indefinitely, each of the sums Sp and S'g must increase indefinitely, for otherwise the series would diverge, or else converge absolutely. On the other hand, since the series is supposed to converge, the general term must approach zero. We may now form a new series whose sum is A in the following manner : Lei OS take positive terms from the given series in the order in which they occur in it until their sum exceeds A. Let us then add to these, in the order in which they occur in the given series, negative terms until the total sum is less than A. Again, beginning with the positive terms where we left off, let us add positive trnms until the total sum is greater than A. We should then return to the Mgative terms, and so on. It is clear that the sum of the first n terms of the new series thus obtained is alternately greater than and then less than A, and that it differs from ^ by a quantity which approaches zero as its limit. IM. Absl's test The following test, due to Abel, enables us to establish the eoBTergenoe of certain series for which the preceding teste fail. The proof is ' apon the lemma sUted and proved in § 76. JM «o + wi -H . . . -t u„ -f • • • VIII, $166] CONSTANT TERMS S49 be a series which conTorgat or which if imMa wfaBlt (that !■• for wUdi Qm na of tiie first n t4;rinH in always leas than a flzed nombtr it In *»ir*^it ▼■!■•). Again, let Co, «i, •••, c., • • be a monotoDically decreasing seqnenoe of poaitiTe nnmbers which apprmeb zero as n becomes infinite. Then the (17) «oMo + *iKi+ •• + <.!«• + ••• converges under the hypothean made abofte. For by the hypotheses made above it foDows that for any value of n and p. Hence, by the lemma jntt referred to, wo may writ* Sinc^ cm+i approaches zero as n becomes infinite, n may be chosen so large that the absolute value of the sum <«+iWi. + i 4- • • • + €,+,u,^.^ will be less than any preassigned positive number for all values of p. Tho series (17) therefore converges by the general thecnvm of | 167. When the series Uo+UiH l-u, + --- reduces to the series 1_ 1 + 1-1 4-1 _ 1 whose terms are alternately +1 and — 1, the thf >rfm -•( tln^ .»:ir i»- r. -....». u> the theorem stated in § 165 with regard to alternating aeries. As an example under the general theorem consider the series 8intf + sin2tf + 8in8* + .- + sinn#-|--. which is convergent or indeterminate. For if sin # s 0, every term of tba aariM is zero, while if sin 6^0, the sum of the first n terms, by a formaU of Trigo- nometry, is equal to the expression sin — ^..(--±1,), 2 which is less than | l/sin {6/2) \ in absolute value. It follows that tba aariaa sin* rin2f ^ •toi*# ^ .. 1 2 » converges for all values of $. It may be shown In a similar manner that the series ooetf ooe2f . ooan# + — — — + • • • + _ ▼ • • • 2 n converges for .ill v.imts vi 4 except $ = 2kw» 860 INFINITE SERIES [VIII, §167 Corollary. Restricting ourselves to convergent series, we may state a more general theorem. Let be a conrergent series, and let be any monotonically increasing or decreasing sequence of positive numbers which approach a limit k different from zero as n increases indefinitely. Then the ieriea (18) «0W0 + «lWl H + enlLn + '" aUo converges. For definiteness let us suppose that the c's always increase. Then we may write eo = k-ao, €i = k — ai, • • •, €„ = A; - a„, ••-, where the numbers ao , o'l , • • • , <3'n , • • • form a sequence of decreasing positive numbers which approach zero as n becomes infinite. It follows that the two series kuo-h kui + hkiLn + • • • , aoUo + rt'iMi H + a„Un H both converge, and therefore the series (18) also converges. II. SERIES OF COMPLEX TERMS MULTIPLE SERIES 167. Definitions. In this section we shall deal with certain gen- eralizations of the idea of an infinite series. Let (19) Uq -{- Ui -\- ut -{ \-u„-\ be a series whose terms are imaginary quantities : «o = ao + *o**> Ui = ai + bii, •••, w„ = a„ + ^„i, Such a series is said to be convergent if the two series formed of the real parts of the successive terms and of the coefficients of the imaginary parts, respectively, both converge: (20) tto + «! + a« + • • • + a„ + • • • = 5', . (21) 6o+*i+ft, H----+^»„ + --- = 5". Let 5' and S" be the sums of the series (20) and (21), respectively. Then the quantity 5 = 5' -f iS" is called the sum of the series (19). It is evident that S is, as before, the limit of the sum S^ of the first n terms of the given series as n becomes infinite. It is evident that a series of complex terms is essentially only a combination of two series of real terms. VIII, §168] COMPLEX TERMS MULTIPLE ftgttrM $5^ When the series of ahtoluU values of the terms of the serim (19) (22) V«! + « + >/«} + ^ + - •+v/5T^.4-. convergesy each of the ser ies (20) a^id (21) evid emilff lutely, for \a,\ S \/«i + *! and \K\ i N/ajV ^. In this case the series (19) is said to be absoltUehj etmverg eni . The sum of such a series is not altered by a change in the order ef the terms f nor by grouping the terms together in any way. Conversely t if each of the series (20) an d (21) e o nv erge e aheoimtely, the series (22) converges absolutely f for v«J + ^ 5 l^*.! + |^«|» Corresponding to every test for the convergence of a feriet of positive terms there exists a test for the absolute oonTWgBOee of any series whatever, real or imaginary. Thus, if the abeeU d e vahie of the ratio of two consecutive terms of a series |t<,^i/i«,|, after a cer- tain term, is less than a fixed number less than unity^ the series eon" verges absolutely. For, let f /", = | m, | . Then, since ( m, ^ , /m. ' < ik < 1 after a certain term, we shall have also %^<A-<1, which shows that the series of absolut*- valii«*s 6^0 + ^i + • • • 4- 1. + • • converges. If |w„^,/w.| approaches a limit I as n becomes if\/lniie, the series converges if 1<1, and diverges if l>\. The first hmif is self-evident. In the second case the general term ic, does not approach zero, and consequently the series (20) and (21) caimoi both be convergent. The case / = 1 remains in doubt. More generally, if « be the greatest hmit of -v^ m « becoiMs infinite, Ike series (19) converges if u<l, and diverges if t0>\. For tn Ibe UU«r CMt Um modulus of the general term doe« not approach «ero (toe 1 161). The CSH la which u = 1 remains in doubt — the aeries msj be abK>latel7 ooavsifMt, iteply convergent, or divergent. 168. Multiplication of aeries. Let (23) Uo + M, -I- ti, 4- •• -f u.-f •••, (24) be any two serit'> .s i ,i; . ; • ^ •:..• !.:•«: series by terms ol ihr >»... ni ^ .• i; » ■■' ■■ i-**" 1' 852 INFINITE SERIES [VIII, §168 together all the products w,Vy for which the sum i-^j of the sub- scripts is the same ; we obtain in this way a new series If each of the series (23) and (24) is absolutely convergent, the series (25) converges, and its sum is the product of the sums of the two given series. This theorem, which is due to Cauchy, was gener- alized by Mertens,* who showed that it still holds if only one of the series (23) and (24) is absolutely convergent and the other is merely convergent. Let us suppose for definiteness that the series (23) converges absolutely, and let w^ be the general term of the series (25): The proposition will be proved if we can show that each of the differences Wf, + Wi + "- + w^^ -(uo^u^ + ...^u„) (v^ 4_ vi 4- . . . 4- v^), ««'0 + «^l + • • • + tV^n+l - (^0 + t^l + • • • + U, + l)(Vo + ^'l 4- • • • 4- V„ + i) approaches zero as n becomes infinite. Since the proof is the same in each case, we shall consider the first difference only. Arranging it according to the w's, it becomes 8 = "o(V. + I -f • • • + V2n) -h ^l ('^„ + l + • • • + f2«-l) + • • ' + i^n-1^ + 1 + W, + l(Vo + • • • + V„-l) + U„^^(Vo + • • • + V„_2) + • • • + ^2,1^0- Since the series (23) converges absolutely, the sum Uo-\-Ui-\ \-U^ is less than a fixed positive number A for all values of n. Like- wise, since the series (24) converges, the absolute value of the sum «^o + t^i H h V. is less than a fixed positive number B. Moreover, corresponding to any preassigned positive number c a number m exists such that for any value of p whatever, provided that n>m. Having so chosen n that all these inequalities are satisfied, an upper limit of the quan- tity |a{ ii given by replacing u^, u„ w„ • . ., m,. by Uo, U^ f/», • •, ^,., • CrtW$ Journal, Vol. LXXIX. VIII, $ 109] COMPLEX TERBiS MULTIPLE SERIES t6i respectively, v,^. » 4- v,^, -»-••• + r,^, by ^/{A + B), and finally of the expressions v© -f vi -f- • • • -f w,_j, v, -f H ».-f ,•••,«»• by JL This gives or |8|<J^(i/o + ^i+-- + t/,-,)-hB(r.n 4- •• + //,.) whence, finally, | S | < c. Hence the difference 5 actually does i4>proach zero as n becomes infinite. 169. Double series. Consider a rectangular network which is lim- ited upward and to the left, but which extends indefinitely down- ward and to the right. The network will contain an infinite number of vertical columns, which we shall number from left to right from to + X . It will also contain an infinite number of horizontal rows, which we shall number from the top downward from to -f oo. Let us now suppose that to each of the rectangles of the network a certain quantity is assigned and written in the corresponding rec- tangle. Let a,t be the quantity which lies in the »th row and in the A;th column. Then we shall have an array of the form (26) «00 aoi «os • • Oo. ••• «.o a,, «ij • • ^l, ••• OlO <h\ «M «-0 ^m\ **-.« - ••., We shall first suppose that each of the elements of this array is real and positive. Now let an infinite sequence of curves Cj, ' . , be drawn across this array as follows : 1) Any one of them forms with the two straight lines which Iwund the array a closed curre which entirely surrounds the preceding one ; 2) The distance from any fixed pdni to any point of the curve C„ which is otherwise entirely arbitrary, becomes infinite with n. Let 5, be the sum of the elemenU of the array which lie entirely inside the closed curve composed of C, and 354 INFINITE SERIES [VIII, § 169 the two straight lines which bound the array. If 5„ approaches a limit 5 as n becomes infinite, we shall say that the double series (27) X X«- 1 = k=0 converges f and that its sum is S. In order to justify this definition, it is necessary to show that the limit 5 is independent of the form of the curves C. Let C{, Cg, •••, Cl,, ••• be another set of curves which recede indefinitely, and let 5/ be the sum of the elements inside the closed curve formed by C{ and the two boundaries. If m be assigned any fixed value, n can always be so chosen that the curve C, lies entirely outside of C'^. Hence S'^< S^, and therefore ^L = Sj for any value of m. Since 5^ increases steadily with m, it must approach a limit S' < 5 as m becomes infinite. In the same way it follows that S < S'. Hence S' = S. For example, the curve C^ may be chosen as the two lines which form with the boundaries of the array a square whose side increases indefinitely with *, or as a straight line equally inclined to the two boundaries. The corresponding sums are, respectively, the following : aoo + (a,oH-aii + aoi)+"-+(a«o + a„i + ----fa„„ + a„-i.„4---- + aon)> aoo+(aioH-aoi)+(a2o + aii + ao2)+---+Ko + a„_i,i + --- + ao«)- If either of these sums approaches a limit as n becomes infinite, the other will also, and the two limits are equal. The array may also be added by rows or by columns. For, sup- pose that the double series (27) converges, and let its sum be S. It is evident that the sum of any finite number of elements of the series cannot exceed S. It follows that each of the series formed of the elements in a single row (28) a,o -}- a.i + • • • + «,n + • • • , i = 0, 1, 2, • • • , converges, for the sum of the first n+1 terms a^Q + a,i H h *•» cannot exceed S and increases steadily with n. Let o-, be the sum of the series formed of the elements in the tth row. Then the new series surely converges. For, let us consider the sum of the terms of the array Sa^^ for which t</>, A;<r. This sum cannot exceed S, and increases steadily with r for any fixed value of jo; hence it approaches a limit as r becomes infinite, and that limit is equal to (^^) <^» + <r, + • • • + CTp I vni,§l«9] COMPLEX TER>f- 'ILTIPLE SERIES 856 for any fixed value of p. It foUow8 that <r« + cti 4- . • 4. ^^ '^^ rr ^ exceed S and increases steadily with p. Consequently the seriet (29) converges, and its sum S is less than or equal to .V. Coov«nely, if eacii of the series (28) converges, and the series (29) oonverget to a sum :^, it is evident that the sum of any finite number of elamente of the array (26) cannot exceed 2. Hence S<X and ooDMqiiaiitlj The argument just given for the series formed from the elements in individual rows evidently holds equally well for the aeriea formed from tiie elements in individual columns. The sum of a ftimwufiMf double series whose elements are all positive may be evalmated ky roivSf by columnsy or by means of curves of any form which rectie indefinitely. In particular^ if the series eowBergea when added by tmoe^ it will surely converge when added by columns^ and the eum will be the same. A number of theorems proved for simple series of positiYe terms may be extended to double series of positive elements. For example : if each of the elements of a double series of positive elewiemte is lesSf respectively, than the correeponding elements of a known eon' vergent double series, the first series is also convergent; and ao forth. A double series of positive terms which is not convergent is said to be divergent. The sum of the elements of the corresponding array which lie inside any closed curve increases beyond all limit as the curve recedes indefinitely in every direction. Let us now consider an array whose elements are not all positire. It is evident that it is unnecessary to consider the cases in which all the elements are negative, or in which only a finite number of elements are either positive or negative, since each of these eases reduces immediately to the preceding case. We shall therafore sup- pose that there are an infinite number of positive elements and an infinite number of negative elements in the array. Let a^ be the general term of this array T. If the array T, of positiTe alemsnta, each of which is the absolute value \a^^\ of the corresponding elsoiant in Ty converges, the array T is said to be absolutely eemperyenL Snob an array has all of the essential properties of a oooTergant array of positive elements. In order to prove this, let us consider two auxiliary arrays T and r\ defined as follows. The array T* is formed from the array T by replacing each negative element by a lero, retaining the positive elements as they stand. Likewise, the array 7^ is obtained from the array T by replacing each positive element by a sero and chang- ing the sign of each negative element Each of the arrays T and 7* 866 INFINITE SERIES [VIII, §169 converges whenever the array Ti converges, for each element of 7^, for example, is less than the corresponding element of Tj. The sum of the terms of the series T which lie inside any closed curve is equal to the difference between the sum of the terms of V which lie inside the same curve and the sum of the terms of r" which lie inside it. Since the two latter sums each approach limits as the curve recedes indefinitely in all directions, the first sum also approaches a limit, and that limit is independent of the form of the boundary curve. This limit is called the sum of the array T. The argument given above for arrays of positive elements shows that the same sum will be obtained by evaluating the array T by rows or by columns. It is now clear that an array whose elements are indiscriminately positive and negative, if it converges absolutely j may be treated as if it were a convergent array of positive terms. But it is essential that the series Tj of positive terms be shown to be convergent If the array Ti diverges, at least one of the arrays T and T' diverges. If only one of them, T for example, diverges, the other T'" being convergent, the sum of the elements of the array T which lie inside a closed curve C becomes infinite as the curve recedes indefinitely in all directions, irrespective of the form of the curve. If both arrays T and T' diverge, the above reasoning •howi only one thing, — that the sum of the elements of the array T inside a closed curve C is equal to the difference between two sums, each of which Increaaes indefinitely as the curve C recedes indefinitely in all directions. It may happen that the sum of the elements of T inside C approach different limits according to the form of the curves C and the manner in which they recede, that is to say, according to the relative rate at which the number of positive terms and the number of negative terms in the sum are made to increase. The sam may even become infinite or approach no limit whatever for certain methods of recession. As a particular case, the sum obtained on evaluating by rows may be entirely different from that obtained on evaluating by columns if the array is not absolutely convergent. The following example is due to Amdt.* Let us consider the array (SI) 2(2/ "sU'sls) "4(4) '.'••' ;(V") "FTi(?Tl)' 2(2} "iW'sls) "4(4)*--'jp(^) -jilfe)* • Orunert'B ArcMv, Vol. XI, p. 319. VIII. fie»] COMPLKX TER1C8 MULTIPLE HER1K8 367 which contains an infinite number of poaicife ao<l aa elementii. Each of the serien foraad from tte alMBMrti in a those in a single column coQTWfia. Tha ram of tha terma in the nth row ia aTtdantlgr i/iV JL Hence, evaluating the array (81) by rowa, the reaoli obuUnad Im aqoal to tha sum of the convergent seriea which ifl 1/2. On the other hand, the aariaa formed from the elemniili In tha (p — l)th column, that is, converges, and its sum is P P+1 P(P+1) P + 1 P* Hence, evaluating the array (31) by columns, the reanlt obtained la equal to the sum of the convergent series (i-i)*(:-i)--(;^-j) which is — 1/2. This example shows clearly that a double aerlee ahonld not be need ia a calculation unless it is absolutely convergent. We shall also meet with double series whose elements are complex quantities. If the elements of the array (26) are complex, two oihar arrays V and T' may be formed where each element of 7* ia th« real part of the corresponding element of Tand each element of 7** is the coefficient of t in the corresponding element of 7*. If the array T^ of absolute values of the elements of T, each of whoae elements is the absolute value of the corresponding element of T, converges, each of the arrays T and T" oonverges abiolntaly, and the given array T is said to be absoluUlif eomn^rgenL The sum of the elements of the array which lie inside a Tariable cloaed cunre approaches a limit as the curve recedes indefinitely in all directions. This limit is independent of the form of the Tariable corre, and it is called the sum of the given array. The sum of any absolutet^ convergent array may also be evaluated by rows or by colimins. 858 INFINITE SERIES [VIII, §170 170. An absolutely convergent double series may be replaced by a simple seriM formed from the same elements. It will be sufficient to show that the roctangles of the network (26) can be numbered in such a way that each rec- tangle ha» a definite number, without exception, different from that of any other roctangle. In other words, we need merely show that the sequence of natural nambeFB (82) 0, 1, 2, ..., n, •.-, and the assemblage of all pairs of positive integers (i, k), where i>0, A;>0, can be paix«d oil in such a way that one and only one number of the sequence (32) will correspond to any given pair (i, fc), and conversely, no number n corresponds to more than one of the pairs (i, A:). Let us write the pairs (i, k) in order as follows : (0,0), (1,0), (0,1), (2,0), (1,1), (0,2), ..., where, in general, all those pairs for which i -h k = n are written down after those for which i + k<n have all been written down, the order in which those of any one set are written being the same as that of the values of i for the various pairs beginning with (n, 0) and going to (0, n). It is evident that any pair (i, k) will be preceded by only & finite number of other pairs. Hence each pair will have a distinct number when the sequence just written down is counted off according to the natural numbers. Suppose that the elements of the absolutely convergent double series SSoifc are written down in the order just determined. Then we shall have an ordinary series (33) ooo + aio + Ooi + aao + an + aoz H + ^no + On-i.i H whose terms coincide with the elements of the given double series. This simple series evidently converges absolutely, and its sum is equal to the sum of the given double series. It is clear that the method we have employed is not the only pos- sible method of transforming the given double series into a simple series, since the order of the terms of the series (33) can be altered at pleasure. Conversely, any absolutely convergent simple series can be transformed into a double series in an infinite variety of ways, and that process constitutes a powerful instrument in the proof of certain identities.* It Lb evident that the concept of double series is not essentially different from that of simple series. In studying absolutely convergent series we found that the order of the terms could be altered at will, and that any finite number of terms could be replaced by their sum without altering the sum of the series. An attempt to generalize this property leads very naturally to the introduction of double series. 171. Multiple series. The notion of double series may be generalized. In the first place we may consider a series of elements a„,„ with two subscripts m and n, each of which may vary from — oo to -|- oo . The elements of such a series may be arranged in the rectangles of a rectangular network which extends indefinitely in all directions ; •Tannery. Tntroduetion a la thtforie des fonctions d'une variable, p. 67. Viii, §172] COMPLEX TERMS M CLT J PLE SERIES S59 ii 1.^ evident that it may be divmea into four double Mriet of tht type we have just studied. A more important generalization is the following. Let ut a series of elements of the type a^,^,,,^ where the tul TKi, TTi^f "t rn^ may take on any values from to + oo, or from — oe to + 00, but may be restricted by certain inequalitiea. Although no such convenient geometrical fonn as that used above if ATailable when the number of subscripts exceeds three, a slight oonsideratioo shows that the theorems proved for double series admit of immediate generalization to multiple series of any order p. Let us first sup- pose that all the elements «„,,.„,,...,,« are real and positive. Let Si be the sum of a certain number of elements of the given seriee, ^ the sum of .s\ and a certain number of terms previously neglected, Ng the sum of .s'2 and further terms, and so on, the successive sums .Si, S^y • •, N„, •• being formed in such a way that any particular elemcnt of the given series occurs in all the smns past a certain one. If S^ approaches a limit S as ti becomes infinite, the given seriee is said to be convergent^ and S is called its sum. As in the case of double series, this limit is independent of the way in which the successive sums are formed. If the elements of the given multiple series have different signs or are complex quantities, the series will still surely converge if the series of absolute values of the terms of the given series ooDTeiget. 172. Generalization of Cauchy's theorem. The following theotem, which is a generalization of Cauchy's theorem (§ 161), enables us to determine in many cases whether a given multiple series is conver- gent or divergent. Let /(a-, y) be a function of the two variablet x and y which is positive for all points (x, y) outside a certain closed curve r, and which steadily diminishes in value as the p« •/) recedes from the origin.* Let us consider the value of t: . io integral///(a;, y) dx dy extended over the ring-shaped region r and a variable curve C outside T, which we shall allow to indefinitely in all directions ; and let us compare it with the double series 2/(m, n), where the subscripts m and n may assume any posi- tive or negative integral values for which the point (w, n) lies out^ side the fixed curve T. Then the doubU mrim oemoergu iftks integral approaches a limit f and convenely. • All that is necessary for the present proof b UuU/(«i, |fi)^/(«^. Wd zi>z, and Vx'^Vt outside T. It Is easy to adapt the proof to rtOl bm hypotheses. — TaAMs. 860 INFINITE SERIES [vm,§m The lines a; = 0, x = ±1, a; = ± 2, • • andy = 0, y = ±1, y = ±2, • •• divide the region between r and C into squares or portions of squares. Selecting from the double series the term which corresponds to that comer of each of these squares which is farthest from the origin, it is evident that the sum 2/(w, n) of these terms will be less than the value of the double integral Jff(xj y) dx dy extended over the region between r and C. If the double integral approaches a limit as C leoedes indefinitely in all directions, it follows that the sum of any number of terms of the series whatever is always less than a fixed number ; hence the series converges. Similarly, if the double series converges, the value of the double integral taken over any finite region is always less than a fixed number; hence the integral approaches a limit. The theorem may be extended to multiple series of any order jo, with suitable hypotheses; in that case the integral of comparison is a multiple integral of order p. As an example consider the double series whose general term is l/(m* -\- w*)**, where the subscripts m and n may assume all integral values from — oo to + oo except the values m = n = 0. This series converges for /ti > 1, and diverges for ^^1. For the double integral <»» ff<^> extended over the region of the plane outside any circle whose center is the origin has a definite value if /a > 1 and becomes infinite if /i<l (§133). More generally the multiple series whose general term is 1 (mJ + mi + '-' + mJ)'*' where the set of values mi = mg = • • • = mp = is excluded, con- verges if 2/A > p.* III. SERIES OF VARIABLE TERMS UNIFORM CONVERGENCE 173. Definition of uniform convergence. A series of the form (35) «o(aj)+ Mi(a;)-f • • + t^n(«)+ •••, whose terms are continuous functions of a variable a; in an inter- ral (a, 6), and which converges for every value of -^ belonging to that interval, does not necessarily represent a continuous function, UMorwna are to be found in Jordan's Cours d 'Analyse, Vol. I, p. 163. VIII, $173] VARIABLE TERMS 861 as we might be tempted to believe. In order to prore the fael W9 need only consider the series studied in | 4 : which satisfies the above conditions, but whose sum is di«oontinuous for X = 0. Since a large number of the functions which occur in mathematics are defined by series, it has been found neceeiary to study the properties of functions given in the form of a series. The first question which arises is precisely that of determining whelber or not the sum of a given series is a continuous function of the variable. Although no general solution of this problem is known, its study has led to the development of the very important notion of uniform converyence. A series of the type (35), each of whose terms is a function of x which is defined in an interval (a, 6), is said to be w»tforml}f eam^ vergent in that interval if it converges for every value of x between a and b^ and if, corresponding to any arbitrarily preassigned positive number e, a positive integer JV, independent of or, can be found such that the absolute value of the remainder R^ of the given aeries ^* = w.+i(-^)+ ".+«(«)+••• + •*.+,(«)+ ••• is less than « for every value of n>N and for every value of » which lies in the interval (a, b). The latter condition is essential in this definition. For any pre- assigned value of x for which the series converges it is apparent from the very definition of convergence that, corresponding to any positive number €, a number N can be found which will satisfy the condition in question. But, in order that the series should con- verge uniformly, it is necessary further that the same number N should satisfy this condition, no matter what value of x be selsoted in the interval {a, b). The following examples show that soeh is not always the case. Thus in the series considered just above we have The series in question is not uniformly convergent in the inter- val (0, 1). For, in order that it should be, it would be «eeesMrjf (though not sufficient) that a number N exist, such that 1 ^ (1-hx*)' 862 INFINITE SERIES [VllI, § 173 for all values of x in the interval (0, 1), or, what amounts to the same thing, that 1 -f a:^ > e^ 1 1 1 Whatever be the values of N and e, there always exist, however, positive values of x which do not satisfy this inequality, since the right-hand side is greater than unity. Again, consider the series defined by the equations 5.(x)=nxe-^, S,{x)=^, u,(x) = S, - S„_,, w = l,2, .... The sum of the first n terms of this series is evidently S,^ (x), which approaches zero as n increases indefinitely. The series is therefore convergent, and the remainder Rn{x) is equal to — wxe""^. In order that the series should be uniformly convergent in the interval (0, 1), it would be necessary and sufficient that, corresponding to any arbi- trarily preassigned positive number c, a positive integer N exist such that for all values of n>N nxe-'^<€, 0<a;<l. But, if a; be replaced by l/?i, the left-hand side of this inequality is equal to e"^/", which is greater than 1/e whenever n> 1. Since c may be chosen less than 1/e, it follows that the given series is not uniformly convergent. The importance of uniformly convergent series rests upon the following property: The sum of a series whose terms are continuous functions of a variable x in an interval (a, h) and which converges uniformly in that intervalf is itself a continuous function ofx in the same interval. Let Xo be a value of x between a and ft, and let Xq-\- hhQ a value in the neighborhood of Xq which also lies between a and b. Let n be chosen so large that the remainder ^«(«) = ^«+i(a5) + w„+2(a;) -h • • • is less than c/3 in absolute value for all values of x in the interval (tf, b), where c is an arbitrarily preassigned positive number. Let/(a;) be the sum of the given convergent series. Then we may write /(x)=^(x)-h/e,(x), where 4i(x) denotes the sum of the first w -f 1 terms, ^{x) = M„(x) 4- w, (x) 4- . . . + w,(x). VllI, §m] VARIAHLE TPUM8 $5$ Subtracting the two equalities /(a^)=0(x,)-fi?.(x,), f{x, 4- A) = ^(x, + A) + i2.(«, 4. A), we find fix, 4- h) ^f(x,) = [^(xo 4- A) - ^(x,)] 4- i?.(x, 4- A) - i?.(x,). The number 71 was so chosen that we have On the other hand, since each of the terms of the series is a oontintt- ous function of x, <f>(x) is itself a continuous function of x. Heoee a positive number rj may be found such that \^x, + h)-4K.x,)\<l whenever \h\ is less than rj. It follows that we shall hare, a fortiori, lA^o-f /o-yi*o)i<3| whenever | ^ | is less than 1;. This shows that f(x) is cootinuoos for x = Xq. Note. It would seem at first very difficult to determine whether or not a given series is uniformly conver^nt in a given inteml. The following theorem enables us to show in many cases that a given series converges uniformly. Let (36) Wo(ar)4- u»(x)4- ••• + «,(«) + be a series each of whose terms is a amHnuoms /knetion qf x in mn interval (a, b), and let (37) 3/,4.3/j4---- + 3r,4--- be a convergent series whose terms are positive oomstamts. Thorn, (/* I M, I < 3/, for all values of x in the interval (a, h) and for ali values of n, the first series (36) eonver^ umtforwdjf in the i m t ero mi considered. For it is evident that we shall have |w. + i + M.^,4----|<iCi^,4-^,^,-»-'-- 864 INFINITE SERIES [VIII, §174 for all values of x between a and h. If JV be chosen so large that the remainder H^ of the second series is less than c for all values of n greater than iV, we shall also have whenever n is greater than iV, for all values of x in the interval (a, h). For example, the series 3/o -h 3f 1 sin x -\- M^ sin 2x H V M^ sin nx -\- •", where 3/o, A/i, 3/,, ••• have the same meaning as above, converges uniformly in any interval whatever. 174. Integration and differentiation of series. Any series of continuous functions which converges uniformly in an inter acU (a, b) may be integrated term by term, provided the limits of integration are finite and lie in the interval (a, b). Let Xq and Xi be any two values of x which lie between a and b, and let jV be a positive integer such that \R^{x)\<€ for all values of X in the interval (a, b) whenever n> N. Let f(x) be the sum of the series f{x) = Uo(x) + wi (x) + • • • + i^„(ic) + • • •, and let. us set ^.= 1 f{x)dx-j Uodx- I uidx / \^dx=( ^R^dx. The absolute value of D, is less than €\xi—Xo\ whenever n>N. Hence Z). approaches zero as n increases indefinitely, and we have the equation / f{x)dx=l Uo(x)dx-{- ui(x)dx + -..i- f \(x)dx + --.. *''• *^*% *>'*o Jx^ Considering x^ as fixed and x^ as variable, we obtain a series J Uo(x)dx + -'.-^ j u^(x)dx + ... which converges uniformly in the interval (a, b) and represents a oondDuout function whose derivative is f{x). VI", §174] VARIABLE TER>IS Conversely, any eanverffent series may be d^emUiaied term hy term if the resulting series converges untformly* For, let f(x) a tio(«) + tt|(x) + •• • + ti.(x) + ... be a series which converges in the interval (a, h). Let us suppose that the series whose terms are the derivatives of the terms of the given series, respectively, converges uniformly in the tame ioterral, and let </>(x) denote the sum of the new series Integrating this series term by term between two limits jr« aod a^ each of which lies between a and 6, we find I <i>{x)dx = [uo(ar) - Uo(Xo)] + [tt|(x) - tti(av)] + or f <l>(x)dx=/(x)^/(x.). This shows that <^(x) is the derivative oi fXz\. Examples. 1) The integral / X cannot be expressed by means of a finite number of elemsntaiy functions. Let us write it as follows: The last integral may be developed in a series which holds for all values of x. For we have 1.2 ■ 1.2.3 ' ■ 1.2 n and this series converges uniformly in the interval from — i? to -f i?, no matter how large R be taken, since the absolute Talus of aoj * It is assumed in the proof also that aaeh tonn of tiM mw (unctioD. The theorem ia true, however, la g wi i ml. ~ TaAim. 866 INFINITE SERIES [viii, §174 term of the series is less than the corresponding term of the con- vergent series It follows that the series obtained by term-by-term integration X 1 X* 1 x^ converges for any value of x and represents a function whose deriva- tive is (^ — l)/x. 2) The perimeter of an ellipse whose major axis is 2a and whose eccentricity is e is equal, by § 112, to the definite integral w S = 4a r* VI - e2sin20d0. The product c^sin^ lies between and ^{<1). Hence the radical is equal to the turn of the series given by the binomial theorem VI - e^sin^^ = 1 - - e^sin*^ e*sin*0 2 2.4 1.3.5-..(2n-3) , . , 2 . 4 . 6 • . . 2n The series on the right converges uniformly, for the absolute value of each of its terms is less than the corresponding term of the convergent series obtained by setting sin^ = 1. Hence the series may be integrated term by term; and since, by § 116, r ^ sin*" <p d<t> Jo 5«.^. 1.3.6...(2n-l) ;r 2 . 4 . 6 . . . 2n 2 shall have w rVl-c«sin«0(i0 = -(l-lc2-Ae*-Ae6 Vo 2 ( 4 64 256 rl.8.5-.-(2n-3n2 ^ -[ 2.4.6...2n ^ ](2n-l)e^"--}. If the eccentricity e is small, a very good approximation to the exact value of the integral is obtained by computing a few terms. Similarly, we may develop the integral r Vl-e"^sin2 0d0 Jo in a series for any value of the upper limit <f>. Finally, the development of Legendre's complete integral of the first kind leads to the formula * r-—!f^ ;{. + I^+ »,.^ ■ I r l.8.6-.-(2»-l) -|' I J, Vl-CIn'* 2} ^4 ^84 ^ *L 2.4.6...2« J + j" VIII. $174] VARIABLK TERMS %$^^ The definition of uniform convergence way be extended to lene* whose terms are functions of several independent variablet. For example, let Wo i^f y) + W| (x, y) 4- • + M, (X, y) + • be a series whose terms are functions of two independent Tarimblet x and y, and let us suppose that this series converges whenever tht point (x, y) lies in a region R bounded by a oloeed eootoor C, The series is said to be uniformly convergent in the regioo R ML corresponding to every positive number c, an integer N can be found such that the absolute value of the remainder R^ is lees than c whenever n is equal to or greater than Nj for every point (;r, y) inside the contour C. It can be shown as above that the sum of such a series is a continuous function of the two variables x and y in this region, provided the terms of the series are all continu- ous in R. The theorem on- term-by-term integration also may be generalixed. If each of the terms of the series is continuous in R and if /(x, y) denotes tlie sum of the series, we shall have \ J f{^iy)dxdy= j j u^{x,y)dxdy'^j j u^{x,y)dxdy '\- -. ^J J ^^n{x>y)dxdy{'"-, where each of the double integrals is extended over the whole Inte- rior of any contour inside of the region R. Again, let us consider a double series whose elements arw tunruoos of one or more variables and which converges absolutely for all seta of values of those variables inside of a certain domain D. Let the elements of the series be arranged in the ordinary rectangular amy, and let R^ denote the sum of the double series outside any eloeed curve C drawn in the plane of the array. Then the given double series is said to converge uniformly in the domain D if oomspood- ing to any preassigneil number c, a closed curve i?, not depaodeni on the values of the variables, can be drawn such that \R,\<t for any curve C whatever lying outside of K and for any set of values of the variables inside the domain D. It is evident that the preceding de6nitions and tbeorams may be extended without difficulty to a multiple series of any order elements are functions of any number of yariablea. 868 INFINITE SERIES [VIII, §176 NoU. If a aeries does not converge uniformly, it is not always allowable to integrate it term by term. For example, let us set fl;(x) = iU5e-'«*, 5o(x) = 0, u^(x) = S„-5„_i. n = l, 2, .... The series whose general term is Un (x) converges, and its sum is^ero, since Sn (x) si^KYWches sere as n becomes infinite. Hence we may write /(x) = = ui(x) + 1I2(X) + '. . + M«(x) + . ., whence Sq/(x) dx = 0. On the other hand, if we integrate the series term by term between the limits zero and unity, we obtain a new series for which the sum of the first n terms is /»1 T-p-nx'-il 1 j;s.(x)d» = -[— ]^=-(l-e-»), which approaches 1/2 as its limit as n becomes infinite. 175. Application to differentiation under the integral sign. The proof of the formula for differentiation under the integral sign given in § 97 is based essentially upon the supposition that the limits Xq and X are finite. If X is infinite, the formula does not always hold. Let us consider, for example, the integral sin ax ax, a > 0, This integral does not depend on a, for if we make the substitu- tion y = ax it becomes [ siny y If we tried to apply the ordinary formula for differentiation to F(a), we should find ^'^=£ F^(a) = j cos ax dx This is surely incorrect, for the left-hand side is zero, while the right-hand side has no definite value. Sufficient conditions may be found for the application of the ordinary formula for differentiation, even when one of the limits is infinite, by connecting the subject with the study of series. Let u« first consider the integral X f(x)dx, whieh we shall suppose to have a determinate value (§ 90). Let •i, a,, ..., a,, .. ■ be an infinite increasing sequence of numbers, all VIII, (m] VARIABLE TBRlfg 359 greater than Oo, where a, becomea inlmite with it. If we aei U.^J\x)dx, U,=J\x)dx, ..., U.^J'^'*/{g)du, the series converges and its sum is /^* *f(^jr) rfar, for the sum A\ of the finrt n termi is equal to f^f(x)dx. It should be noticed that the converse is not always true. If, for example, we set /(x) = C08x, 00=0, ai = 7r, «. = »«•, -, we shall have %/nn COS xdx ssO, Hence the series converges, whereas the integral f'ooBxdx ap- proaches no limit whatever as / becomes infinite. Now let f{x, a) be a function of the two variables x and a which is continuous whenever x is equal to or greater than <^ and a lies in an interval (a©, ai). If the integral J'/{x, a)dx approaches a limit as I becomes infinite, for any value of a, that limit is a function of a, FL.\=\ /{x,a)dx, '\ which may be replaced, as we have just shown, by the sum of a convergent series whose terms are continuous functions of a: F(a) = Uo(a) + f/^ (a) -f • • 4- U,(a) + ••., This function F(a) is continuous whenever the series copf w g M ani- formly. By analogy we shall say that ths integral {^* J{x^ a) dx converges uniformly in the interval (a©, <ti) if, oorrespoiidiiig to any preassigned positive quantity c, a number ^ independent of a emu be found such that | Jj"^*/!;*, a)dx\<% whenever /> S, for a^y value of a which lies in the interval («•, at).* If tlie integnd oonwfti • See W. F. Osgood. AwmU <(f M ath sm o Hm , M Mtkt, Vol. HI 0«H). F im — Trams. 370 INFINITE SERIES [VIII, §176 unifonnly, the series will also. For if a„ be taken greater than N, we shall have I f(x,a)dx <€; henoe the function F{a) is continuous in this case throughout the intenral (a©, ai). Let us now suppose that the derivative df/da is a continuous function of x and a when x^a© and aQ<a^aij that the integral X f-dx Ca has a finite value for every value of a in the interval (aro, o-j), and that the integral converges uniformly in that interval. The integral in question may be replaced by the sum of the series X where The new series converges uniformly, and its terms are equal to the corresponding terms of the preceding series. Hence, by the theorem proved above for the differentiation of series, we may write n<^)=£'Yj^. In other words, the formula for differentiation under the integral sign still holds, provided that the integral on the right converges uniformly. The formula for integration under the integral sign (§ 123) also may be extended to the case in which one of the limits becomes infinite. Let /(a*, a) be a continuous function of the two variables X and a, for x>aQ,aQ<a<a^. If the integral S^"^ f{x, a) dx is uni- formly convergent in the interval (ao, «i), we shall have I dx f f(x,a)da= f da I f(x,a)dx. To prove this, let us first select a number l> a^^ then we shall hare (B) J ^f '/(«' a)da = j 'da Cf(ic, a) dx , VIII. §176] VARIABLE TERMS S7l As / increases indefinitely the right-hand side of tkii approaches the double integral da j JXx,a)dXt ioi the difference between these two double integrmlB is equal to f{z, a)dx. SH Suppose N chosen so large that the absolute value of the integral ii^^fi^y oc)dx is less than c whenever I is greater than A', for any value of a in the interval (ao, a^). Then the absolute value of the difference in question will be less than c|ai — a^\^ and therefore it will approach zero as I increases indefinitely. Henoe the left-hand side of the equation (B) also approaches a limit as / becomes in6- nite, and this limit is represented by the symbol J /»■!-• /»«, I dx \ f{x,a)da. This gives the formula (A) which was to be proved.* 176. Examples. 1) Let us return to the int^ral of f 01 : sinz '""=r X "' where a is positive. The integral -J r-^sinxdz, • The formula for difTerentiation may be deduced easily tram t^ foraiala (A). Fbr. suppose that the two functions /(x, a) aiid/«(z. a) are contlBOoas for a^<a<a\, a; > ao ; that the two integrals F(a) = j^ *f{z, a)d*noA ♦(a) = ^ "/, (». a) d» hare finite values ; and that the latter conrergee oniformly in the loU«^ (a^, aj). Froai the formula (A), if (r lies in the intenral (aoi cci), we have where for distinctness a has been replaced by « under tba lalsimU liffa. Mt lUs formula may be written In the form f'*{u) dtt = r * */(x, a)dx-' f* '/(«, a«)ds « tia) - r(a^ , whence, taking the derivattre of each side with raspeol to a, wa ted rCa) = ♦(«). 372 INFINITE SERIES [VIII, §176 obtained by differentiating under the integral sign with respect to a, converges uniformly for all values of a greater than an arbitrary positive number k. For we have U + oc I /»+<» \ e-«*sinxdx </ e-'^'^dx = —er^i , Mid hence the absolute value of the integral on the left will be less than e for all values of a greater than k,ifl> N^ where N is chosen so large that A:e*^ > 1/e It follows that e-'^^sinxdlx. "«') = [- The indefinite integral was calculated in § 119 and gives ■e-<»'*(cosa; + arsinx)l+* _ —1 l+a:2 Jo ~ l+a^' whence we find F{a) = C — arc tan a, and the constant C may be determined by noting that the definite integral F{a) approaches zero as a becomes infinite. Hence C = 7r/2, and we finally find the formula X sinx , ,1 e- «* dx = arc tan — . z a This formula is established only for positive values of a, but we saw in § 91 that the left-hand side is the sum of an alternating series whose remainder E„ is always less than 1/n. Hence the series converges uniformly, and the integral is a con- tinuous function of a, even for a = 0. As a approaches zero we shall have in the limit r /on\ I Sm X , 7t (8») I ^'^=2 2) If in the formula Jo 2 of { 184 we set X = y Va, where a is positive, we find (40) , J^"'V«v.dy = :^a-i, and it is easy to show that all the integrals derived from this one by successive differentiations with respect to the parameter a converge uniformly, provided that a is always greater than a certain positive constant k. From the preceding formula we may deduce the values of a whole series of integrals : («) r t/o V^e-^^dy^l^V^a-i. y2 e-''i^dy = ^ a' 22 f;'^.e-^,y = hl-^.^^^I^^,„-'-^. VIU, Eu] KXERCI8ES 171 By combining iheHe an infinite number of other totafimla may bt •valaated. We have, for example, All the integralB on the right have been evalaated above, and we IIimI 1 fx (2/3)« Va a- 1 Jo 2 \a 1.2 2 2 +(-iy 1.2.3. -^n 2 or, simplifying, 2 Ircr EXERCISES 1. Derive the formula «■ V* ''3.s...(«»~i) ^.iyi 1 il [X- (logx)-] = 1 + Si log* + ;% (logz)« + . . . + r-p— (>«««)•• 1.2..ndx''"'°'" " 1.2^^' I.2...« where iSp denotes the sum of the products of the first n natural numbers taken f at a time. [Muamv.) [Start with the formula and differentiate n times with respect to x.] 2. Calculate the value of the definite integral X '0 by means of the formula for differentiation onder Um Inlafral sign. 3. Derive the formula [First show that dl/da = - 2/.] 374 INFINITE SERIES [Viii, Ex& 4. Derive the formula Jo va by making use of the preceding exercise. 6. From the relation derive the formula 1 1 /* + * a« 2 Jo CHAPTER IX POWER SERIES TRIGONOMETRIC SERIES In this chapter we shall study two particularly important of series — power series and trigonometric serien. Although we ihall speak of real variables only, the arguments used in the study of power series are applicable without change to the case where the variables are complex quantities, by simply substituting the expree- sion inodulHn or absolute value {pt a complex variable) for the ex p ree* 8 ion absolute value (of a real variable).* T. POWER SERIES OF A SINGLh >AiuA»LE 177. Interval of convergence. Let us first consider a series of the form (1) ^lo 4- A^X + A^X} + • • • + vl.-Y- + • ••, where the coefficients >lo> ^i, ^«, ••• are all positive, mod where the independent variable A' is assigned only positive values. It it evident that each of the terms increases with -V. Hence, if the series converges for any particular value of A, say A'l , it oonvergei afoHiori for any value of X less than A'j. Conversely, if the series diverges for the value A',, it surely diverges for any vmlue of X greater than A,. We shall distinguish tlie following cas e s . 1) The series (1) may converge for any value of A whatever. Such is the case, for example, for the series Y Y* A" 1 ^1.2 ' 1.2 -n 2) The series (1) may diverge for any value of X except X — The following series, for example, has this property: l + A+1.2A«-h...+1.2.3..fiA»+ . 3) Finally, let us suppose that the series oonvergt« f«.r r«»rtAio values of A and diverges for other values. Let A, be a valuo of X for which it converges, and let Ag ^ » ^*lu« 'o' which it diverges. •See Vol. \l,^WMa^.^'tm^Jn, 876 376 SPECIAL SERIES [IX, §177 From the remark made above, it follows that Xi is less than Xg. The series converges if A'<A'i, and it diverges if A>Aa. The only uncertainty is about tlie values of X between Aj and X^ . But all the values of A for which the series converges are less than A2, and hence they have an upper limit, which we shall call R. Since all the values of A for which the series diverges are greater than any value of A for which it converges, the number R is also the lower limit of the values of A for which the series diverges. Hence the series (1) diverges for all values of X greater than Ry and converges for all values of X less than R. It may either converge or diverge when X = R. For example, the series l + A + A2-|-...+A» + ..- converges if A < 1, and diverges if A^ ^ 1. In this case R =1. This third case may be said to include the other two by suppos- ing that R may be zero or may become infinite. Let us now consider a power series, i.e. a series of the form (2) tto + aix + a^x^ -\ 1- «„«» -\ , where the coeflScients a, and the variable x may have any real values whatever. From now on we shall set .4,- = |a,.|, Z = |a;|. Then the series (1) is the series of absolute values of the terms of the series (2). Let R be the number defined above for the series (1). Then the series (2) evidently converges absolutely for any value of x between — R and -f- R, by the very definition of the number R. It remains to be shown that the series (2) diverges for any value of x whose absolute value exceeds R. This follows immediately from a funda- mental theorem due to Abel : * If the series (2) converges for any particular value Xq, it converges ahsolutely for any values ofx whose absolute value is less than \xq\. In order to prove this theorem, let us suppose that the series (2) converges for x = Xq, and let 3/ be a positive number greater than the absolute value of any term of the series for that value of x. Then we sliall have, for any value of n, md we may write i4,A* ^•|''Kifi)"<"fe)" • ifecA«rcA« «tir /o «fr<a 1 + ^ X + ^^?^^^^?-lH xa + IX, §177] POWER 8ERIB8 177 It follows that the series (1) conyerges whenerer X<\a^\, whlah proves the theorem. In other words, if the series (2) converges for z « z«, the seriat (1) of absolute values converges whenever X is lest than Ijb^I. Heaoo I^Tol caiinot exceed R, for R was supposed to be the upper limit of the values of X for which the series (1) converges. To sum up, given a power series (2) whose ooefficients maj ha¥« either sign, there exists a positive number R which has the follow- ing properties : The series (2) converges absolutely for any value of x between — R and -f- R, and diverges for any value of x whose nhmluie value exceeds R. The interval (— Rj -^ R) is called the iniervai of convergence. This interval extends from — oo to + oo in the case in which R is conceived to have become infinite, and reduoet to the origin ii R = 0. The latter case will be neglected in what follows. The preceding demonstration gives us no information about what happens when x = R or x = — R. The series (2) maj be absolutalj convergent, simply convergent, or divergent For example, /? = 1 for each of the three series l-\-x -f-x« + ... + x" + ..-, X X 3?* ! + _ + _ + . .. + _ + ..., for the ratio of any term to the preceding approaches x as its limit in each case. The first series diverges for z = ± 1. The seoood series diverges for x = 1, and converges for x = — 1. The third con- verges absolutely for x = ±1. Note. The statement of AbePs theorem may be made more geoaial, for it is sufiicient for the argument that the absolute value of aoj term of the series aoH-«i«oH + «,*oH be less than a fixed number. Whenever this condition is ilHlil e d, the series (2) converges absolutely for any value of x whose aboolnte value is less than |xo|. The number R is connected in a very simple wmy with the nnmbar w dsteed in § 160, which is the greatest limit of the eeqaenoe For if we coosider the analogous sequence AiX, VATTi <a7X», .A^X- 378 SPECIAL SERIES [IX, §178 it Lb evident that the greatest limit of the terms of the new sequence is wX. The sequence (1) therefore converges if X < 1/w, and diverges if X > 1/w ; hence « = !/•».• 178. Continuity of a power seriea. Let f(x) be the sum of a power series which converges in the interval from — R to -{- R, (3) fix) = ao + «!« + • • • + a„«'» + • • • , and let /?' be a positive number less than R. We shall first show that the series (3) converges uniformly in the interval from — R' to -I- R'. For, if the absolute value of x is less than R', the remainder R^ of the series (3) is less in absolute value than the remainder of the corresponding series (1). But the series (1) converges for A' = R', since R' < R. Consequently a number N may be found such that the latter remainder will be less than any preassigned positive number c whenever n'tN. Hence | i2„ | < c whenever n^N provided that \x\<R\ It follows that the sum f(x) of the given series is a continuous function of x for all values of x between — R and + R. For, let Xq be any number whose absolute value is less than R. It is. evident that a number R' may be found which is less than R and greater than \xq\. Then the series converges uniformly in the interval (— R', + R'), as we have just seen, and hence the sum/(cc) of the series is continuous for the value Xq, since Xq belongs to the interval in question. This proof does not apply to the end points + R and — R of the interval of convergence. The function f(x) remains continuous, however, provided that the series converges for those values. Indeed, Abel showed that if the series (3) converges for x = R, its turn for z = Risthe limit which the sumf(x) of the series approaches OM X approaches R through values less than R.\ L«t S be the sum of the convergent series 5 = a„ 4- «! /i 4- aai?^ -I- • • • + a^R'* H , • Thii theorem waa proved by Cauchy in his Coura d 'Analyse. It was rediscovered by Hadamard in hie thesis. t As stated above, these theorems can be immediately generalized to the case of series of imaglQary terms. In tliiH case, liowover, care is necessary in formulating the aenemllzaUon. See Vol. 11, § 2tJ«. — Trans. IX. §17H] POWER SKRIE8 879 and let n be a positive mteger such that any one of the tiff is less than a p reassigned positive number c If we set x « R$, and then let $ increase from to 1, a; will increase from to A, and we shall have f(x) =f{eR) = Oo + aid/2 + a,^/?« + • •• + «.^i?- 4- • • •. If n be chosen as above, we may write (S -f(x) = a^R(l-e) + a, /?«(!- ^ + ... + a./?-(l - #^ (4) I 4- «,+ii2-*» 4- ••• -h a.^,/f-^' 4- • •• i - a. + i^"**^""* a.+^^-^'Te--^' , and the absolute value of the sum of the series in the second line can- not exceed e. On the other hand, the numbers ^**, ^"^ *,•••, ^♦^ form a decreasing sequence. Hence, by Abel^s lemma proved in f 75, we shall have In-t-l 'n+1 + i/>n+i + . . . + a.^.^d--^"/?-*"! <0^*U< C. It follows that the absolute value of the sum of the series in the third line cannot exceed c. Finally, the first line of the right-hand side of the equation (4) is a polynomial of degree t» in which vanishes when ^ = 1. Therefore another positive number iy may be found such that the absolute value of this polynomial is less than < whenever 6 lies between 1 — t; and unity. Hence for all such values of we shall have |s-/i;x)|<3,. But € is an arbitrarily preassigned positive number. Hence J{x) approaches .S as its limit as x approaches /?. In a similar manner it may be shown that if the series (3) eon- verges for a; = — 72, the sum of the series for x = — 7? is equal to the limit which /(a;) approaches as x approaches — R through valoei greater than — R. Indeed, if we replace x by — x, this case redoeei to the preceding. ' An application. This theorem enables OS to oomplsle the rtmlts of f 168 regarding the luuliiplication of series. Let (5) S =Uo4i«i + m + . •. + «• + •••. (6) 5' = i»o4»i4»t4--+».+ -- be two convergent series, neither of which convergM absolutely. The series (7) WoOo + (uoti -4- U,t>o) + • • • + (uo». + • • • + «•••) + • • • 880 SPECIAL SERIES [IX, §179 in«y converge or diverge. If It converges, its sum S is equal to the product of the sums of the two given series, i.e. S = SS\ For, let us consider the three power series /(X) = Mo + ttiX + . . • + U,X» + • • •, ^(x) = ro + ti« + •• + »««" + ••• » f (Z) = UoVo + (Mot^i + UiVo)X + • • • + (Mot>» +••• + M„Vo) «" + ••• . Each of these series converges, by hypothesis, when x = 1. Hence each of them converges absolutely for any value of x between — 1 and + 1. For any such Talue of X Cauchy*s theorem regarding the multiplication of series applies and gives us the equation (8) /(x)0(x) = ^(x). By AbePs theorem, as x approaches unity the three functions /(x), ^(x), ^(x) approach S, S\ and 2, respectively. Since the two sides of the equation (8) meanwhile remain equal, we shall have, in the limit, Z = SS\ The theorem remains true for series whose terms are imaginary, and the proof follows precisely the same lines. 179. Successive derivatives of a power series. If a power series f(x) = ^0 + <^i^ 4- cLi^^ H h a„a^" H which converges in the interval (— Rj -\- R) he differentiated term by term, the resulting power series •(9) ai-\-2a2X-\ + 7ia„a;"-^^H converges in the same interval. In order to prove this, it will be sufficient to show that the series of absolute values of the terms of the new series, ^1 + 2^2^ + • • • + nA„X-' + . . ., where i4, = |a^| and X = \x\, converges for X<R and diverges for X>R. For the first part let us suppose that X<R, and let /?' be a num- ber between Xmd R, X<R'<R. Then the auxiliary series R'^ R' R'^ R'Kr'J ^""^R'Kr'J ^"' converges, for the ratio of any term to the preceding approaches X/R't which is less than unity. Multiplying the successive terms of this series, respectively, by the factors IX, $179] POWER SERIES SS| each of which is less than a certaia fixed number, since H'<H,w9 obtain a new series which also evidently conTerges. The proof of the second part is similar to the above. If the seriee where A'j is greater than if, were convergent, the series AiXi-\- 2AtX\ 4- • • • + «i4. AJ 4- • • • would converge also, and consequently the series lA^X^ would con- verge, since each of its terms is less than the corre8|>onding term of the preceding series. Then R would not be the upper limit of the values of A' for which the series (1) converges. The sum /, (x) of the series (9) is therefore a continuous function of the variable x inside the same interval. Since this series con- verges uniformly in any interval (— 72', + /?'), where R*< R,fi(x) is the derivative of f(x) throughout such an interval, by f 174. Since R' may be chosen as near i2 as we please, we may aaeert that the function /(a;) possesses a derivative for any value of x between — R and + Ry and that that derivative is represented by the teriet obtained by differentiating the given series term by term : • (10) f{x) = ax 4- 2a^x -f ... 4- na^x!''' 4- •••. Repeating the above reasoning for the series (10), we see thMi/{x) has a second derivative, /"(x) = 2a, 4- 6a,x 4. . . . 4- n(n - l)o.jf-» 4- •• •, and so forth. The function /{x) possesses an unlimited sequence of derivatives for any value of x inside the interval (— /?, 4- ^)» *n<i these derivatives are represented by the series obtained by differen- tiating the given series successively term by term : (11) f<^\x) = 1 . 2 . . . na, 4- 2 . 3 . . . n(ii 4- 1)«, ♦ I ar 4- • . If we set jr = in these formula, we find ao=/(0), ai^AO), «. = ^' or, in general, 1.2 • Although the oorreepondlng theorem to true for eeriee of li _ proof follows somewhat different linee. See VoL H, 1 3«. — TaAwe. 382 SPECIAL SERIES [IX, §179 The development oif(x) thus obtained is identical with the develop- ment g^ven by Maclaurin's formula : A')=AO) + ifXO) + ~r(0) + --- + ^-^/<'XO) + .... The coefficients a^, a^ •••, a^y ••• are equal, except for certain numerical factors, to the values of the function f(x) and its succes- sive derivatives for x = 0. It follows that no function can have two distinct developments in power series. Similarly, if a power series be integrated term by term, a new power series is obtained which has an arbitrary constant term and which converges in the same interval as the given series, the given series being the derivative of the new series. If we integrate again, we obtain a third series whose first two terms are arbitrary ; and so forth. Examples. 1) The geometrical progression 1-x + x^-x* + "■ + (- 1)" a;" + • • •, whose ratio is — x, converges for every value of x between — 1 and 4- 1, and its sum is 1/(1 + x). Integrating it term by term between the limits and x, where |5c| < 1, we obtain again the development of log(l -I- x) found in § 49 : log(l + =») = MV I'-... + (-!)» gi4-.... This formula holds also for a; = 1, for the series on the right con- verges when 05 = 1. 2) For any value of x between — 1 and + 1 we may write Integrating this series term by term between the limits and x, where |x|<l, we find X X* a;* ^n + i Since the new series converges for aj = 1, it follows that f=i-|^g-7+-+(-i)"2-;n:i+- IX. $m] POWKU SEHiES S8t 3) Let F(x) be the sum uf the convergent aeriee F(x)^l-hjx+ ^^^ / j^^... + ^ 1.2- ./> ^ •^'>" •> where m is any number whatever and |x| < 1. Then we shmll have r(x) = m|_l + -p-x4.- + ^ li...\p-^) ' -'-'-^ -J- Let lis multiply ea(;h side by (1 4- x) and then collect me lerms m like powers of x. Using the identity 1.2 •.(/>-l) 1.2 ../» 1.2. /> which is easily verified, we find the formula (1 + x)F'(*) = m [l + ^ , + !2i^i-|l) ^ + . . . r»(m-l)...(m-p + l) 1 + 1.2. -p '^* J or From this result we find, successively, F'(x) ^ m F(x) "l + a:* log [F(x)] = m log (1 + x) + log r, or F(x)=C(l + x)- To determine the constant C we need merely notice that F(P)^l. Hence C = 1. This gives the development of (1 + x)* found in § fiO : ^ . m , , m(m — 1) • • • (m — » + 1) . , (l4-ar)'»=H- jx-f--- + -^ j 2...P '*' + •••. 4) Replacmg x by - x* and m by - 1/2 in the last formula abovt^ we find y/lZT^^'^ ^ ^'^ 2.4.6... 2ii This formula holds for any value of x between - 1 and -f 1 Inte- grating both sides between the limits and «, whare we obtain the following development for the arcsine arc sin x.lx» 1.3x»^ . 1.8.5 .(2ii-l; ^ ' , inx = j4-23+2746"^ *"*■ 2.4.6..2ii 2ii+l^ ' 884 SPECIAL SERIES [IX, § 180 180. Extension of Taylor's series. Let /(a:) be the sum of a power series which converges in the interval (— R, + R), x^ a point inside that interval, and x^ + h another point of the same interval such that |av,| +1 A| < ii. The series whose sum is /{x^ -}- A), «o + ai(«o 4- A) + <h{x^ + A)' + • • • + a„(«o + hY + "'y may be replaced by the double series obtained by developing each of the powers of {xq + h) and writing the terms in the same power of h upon the same line : (12) This double series converges absolutely. For if each of its terms be replaced by its absolute value, a new double series of positive terms is obtained: ao + «i«o4- ajicj +• •+ a.x^o +••• -\-aih -h2a^Xoh-{'- • + n a^x^-^h -\ + a,A« +• • + ^^^«X-'A^ + - + • (13) n A,\x^\-^\h\ +. H- If we add the elements in any one column, we obtain a series which converges, since we have supposed that | iCo | + 1 A | < /?. Hence the array (12) may be summed by rows or by columns. Taking the sums of the columns, we obtain f{xQ-\-h). Taking the sums of the rows, the resulting series is arranged according to powers of A, and the coefficients of h, A^^ ... are f\x^),f%x^)/2\,'--, respec- tively. Hence we may write (14) f(x. + h) =/(Xo) -f \f\x,) + . . . + j-ilL_^n)(^^) + . . .^ If we assume that |A|<i2-|xJ. This formula surely holds inside the interval from x^ - R -[■\xq\ to a!o + ^-|a?o|» but it may happen that the series on the right converges in a larger interval. As an example consider the function IX. § 180] POWER SERIES $^ (l+ar)"*, where m U not a positiye integtr. The dtfrelopoMiil according to powers of x holda for all values of x between ~ 1 and + 1. Let Xq be a value of x which lies in that interval. Than we may write (1 + «)-= (1 4- a^ 4- X - x,)- = (1 + «,)-(! + «)-, where « = -2. We may now develop (1 -f «)"* according to powers of s, and this new development will hold whenever |«| < 1, Le. for all valaes of x between — 1 and 1 + 2aro. If Xo is positive, the new interval will be larger than the former interval (—1, -f 1). Hence the new formula enables us to calculate the values of the function for values of the variable which lie outside the original interval. Further inveetig»» tion of this remark leads to an extremely important notion, — that of analytic extension. We shall consider this Subject in the sefwod volume. Note. It is evident that the theorems proved for series arranged according to positive powers of a variable x may be extended ately to series arranged according to positive powers of x — «, more generally still, to series arranged according to poeitive of any continuous function ^(x) whatever. We need only consider them as composite functions, ^(x) being the auxiliary fnnctioiL Thus a series arranged according to positive powers of l/x eon- verges foT all values of x which exceed a certain positive constant in absolute value, and it represents a continu ous fun ction of x for all such values of the variable. The function Vx* — a, for example, may be written in the form ± x(l — a/x^*. The expression (1 — e/*^* may be developed according to powers of l/x* for all values of • which exceed Va in absolute value. This gives the formula Vx a-x 2" 2.4a,. • • 2.A,6 2p x»' which constitutes a valid development of Vx* — a whenever x > v«. When X < — V a, the same series converges and represents the func- tion — Vx* - a. This formula may be used advantageously to obCaio a development for the square root of an integer whenever the fifft perfect sijuare which exceeds that integer it known. 386 SPECIAL SERIES [IX, § 181 181. Dominant functions. The theorems proved above establish a close analogy between polynomials and power series. Let (—r, -\-r) be the least of the intervals of convergence of several given power series /i (j*), /a (x), •••,/„ (a:). When |a:|<r, each of these series converges absolutely, and they may be added or multiplied together by the ordinary rules for polynomials. In general, any integral poly- nomial in fi (x)y fi (x), • • • , /n (x) may be developed in a convergent power series in the same interval. For purposes of generalization we shall now define certain expres- sions which will be useful in what follows. Let f(x) be a power series f(x) = tto + dix + a^x^ H h a„a;" ■] , and let ^(x) be another power series with positive coefficients ^(a;) = n'o + «ix + oTgX^H h «'„x" H which converges in a suitable interval. Then the function <f>(x') is said to dominate * the function f{x) if each of the coefficients a„ is greater than the absolute value of the corresponding coefficient of |ao|<«'oj |ai|<«ij •••, |a«|<ar„, •••. Poincar^ has proposed the notation f{x)<4>(x) to express the relation which exists between the two functions f{x) and <^(x). The utility of these dominant functions is based upon the fol- lowing fact, which is an immediate consequence of the definition. Let P(aQy «!, •••, a„) be a polynomial in the first w +1 coefficients of f{x) whose coefficients are all real and positive. If the quanti- ties ao» *i» •••> o,n be replaced by the corresponding coefficients of ^x), it is clear that we shall have For instance, if the function <j>{x) dominates the function /(x), the series which represents \_<^{x)Y will dominate [/(x)]^ and so on. In general, i<i>{x)Y will dominate [/(x)]«. Similarly, if <^ and ^1 are dominant functions for / and /^ , respectively, the product </)<^i will dominate the product ^i; and so forth. •Thl« expremion will be used as a translation of the phrase " 4>{z) est majorante poor la fomiIon/(z)." Likewise, "dominant functions " will be used for " fonctions inajorauuoM." — Trams. IX,} 181] POWER 8KRIK8 $87 Given a power seriee/ljx) which converges in an intenral (— J?, + JT), the problem of determining a dominant function is of ooane iodeter- minate. Hut it is convenient in what follows to make the domi- nant function as simple as possible. Let r be any number lass *hv n R and arbitrarily near R. Since the given series converges for jr « r, the absolute value of its terms will have an upper limit which w« shall call M, Then we may write, for any value of f», A,f*<M or |«.| = ^.<~ Hence the series r whose general term is M(x*/t*)f dominates the given function /l[x). This is the dominant function most frequently used. If the series f(x) contains no constant term, the function r may be taken as a dominant function. It is evident that r may be assigned any value less than R, and that M decreases, in general, with r. But Af can never be less than Aq. If .lo is not zero, a number p less than R can always be found such that the function i4o/(l — x/p) dominates the function /^x). For, let the series 3/ -f- . V - + .1/ ^ + . . • -f- 3/ ^ 4- • • • , where M > A^^ be a first dominant function. If p be a number less than vAq/M and n > 1, we shall have ,p.| = |a.^|x(e)"<«?(?)" whence |a^p"| < vIq. On the other hand, |a,|=3^,. Hence the X X^ jpB Ao + A,--^ A.j^-h •" + A.-A- dominates the function f(x). We shall make use of this fact pree» ently. More generally still, any number whatever which is gieater than or equal to A^ may be used in plaoe of M. 888 SPECIAL SERIES [IX, §182 It may be shown in a similar manner that if Aq = 0^ the function 18 a dominant function, where /i is any positive number whatever. Note, The knowledge of a geometrical progression which dominates the fanc- tion /(x) also enables us to estimate the error made in replacing the function /{z) by the sum of the first n + 1 terms of the series. If the series M/{1 — x/r) dominates /(x), it is evident that the remainder of the given series is less in absolute value than the corresponding remainder of the dominant series. It follows that the error in question will be less than ,(•)■ M 1-5 182. Substitution of one series in another. Let (16) z =f(y) = tto + aiy + • • • + a„2/" + • • • be a series arranged according to powers of a variable y which con- verges whenever \y\<R. Again let (16) y = <f>(x) = b^-\-b,x + ...^ b^x^ + . . . be another series, which converges in the interval (— r, + r). If y» y\ y*, • • • in the series (15) be replaced by their developments in series arranged according to powers of x from (16), a double series <h + axho + a^bl +•••+ aj-. +... (17) . + «i*ia^ + 2aa b^b^x + • • • + na^b^-^b^x -\ 4- a^b^x^ + a^(b\ -h 2b^b^)x^ -f is obtained. We shall now investigate the conditions under which this double series converges absolutely. In the first place, it is necessary that the series written in the first row, IX, § 182] POWER SERIES SS9 should converge abtolntely , i.e. that | b^ \ should be leM than R.^ This eondition is also mffieient. For if it is satisfied, the funotioD 4(x) will be dominated by an expression of the form m/{l — x/p)^ whaie m is any positive number greater than |6,| and where p<r. We may therefore suppose that m is less than R. Let R' be aaolber positive number which lies between m and R. Then the funetioo f(y) is dominated by an expression of the form R' If y be replaced by m/(l — x/p) in this last series, and the powera of 1/ be developed according to increasing powers of x by the binomial theorem, a new double series (18) is obtained, each of whose cociiicuMiis r than the absolute value of the corresponding vv (17), since each of the coefficients in (17) is formed from the coeflioienta <^oy <'^if f^if " f Ki hi ^if ' • ' ^y means of additions and multiplicattoos only. The double series (17) therefore converges abeolutelj pro- vided the double series (18) converges absolutely. If x be replaoed by its absolute value in the series (18), a necessary condition for abeo- lute convergence is that each of the series formed of the terms in any one column should converge, i.e. that |x| < ^ If this oonditioQ be satisfied, the sum of the terms in the (n + l)th column is equal to M _«■(!- f)J Then a further necessary condition is that we shoald luive (19) \'^<f'{^'-l)' or • The case in which the series (15) conrerget for9»ir(M«fl77)wmh» In what follows. — Trans. 390 SPECIAL SERIES [IX, § 182 Since this latter condition includes the former, |a;| <p, it follows that it is a necessary and sufficient condition for the absolute con- vergence of the double series (18). The double series (17) will therefore converge absolutely for values of x which satisfy the inequality (19). It is to be noticed that the series </>(x) converges for all these values of x, and that the corresponding value of y is less than 72' in absolute value. For the inequalities 9 necessitate the inequality |<^(a;)| < iJ'. Taking the sum of the series (17) by columns, we find ao 4- a^<^{x) + a^{^^{x)J + • • • + a„[<^(«)]'' + • • •, that i8,/[<^(J7)]. On the other hand, adding by rows, we obtain a series arranged according to powers of x. Hence we may write (20) /[<^(a;)] = c, + c^x + c^x?- + . . . -f c^x- + • • ., where the coefficients c^, Cj , Cg, • • • are given by the formulae (21) f Co = ao + ai^»o + tta^ -\ f- aj)l -\ , which are easily verified. The formula (20) has been established only for values of x which satisfy the inequality (19), but the latter merely gives an under limit of the size of the interval in which the formula holds. It may be valid in a much larger interval. This raises a question whose solution requires a knowledge of functions of a complex variable. We shall return to it later. Special cases. 1) Since the number W which occurs in (19) may be taken as near R as we please, the formula (20) holds whenever x satisfies the inequality \x\<p(l- m/R). Hence, if the series (15) converges for any value of y whatever, R may be thought of as infinite, p may be taken as near r as we please, and the formula (20) applies whenever |x| < r, that is, in the same interval in which the series (16) converges. In particular, if the series (16) converges for all values of x, and (16) converges for all values of y, the formula (20) is valid for all values of x. IX. § 182] POWER SERIES 891 2) When the constant term b^ of the series (16) is nra^ tbm fun^ tion <f>(x) is dominated by an expression of the form m m, ■-I where p <r and where m is any positive number whatever. An arguinent similar to that used in the general case shows that the formula (20) holds in this case whenever x satisfies the inequality (22) I'KpFT^' where /^' is as near to /? as we please. The corresponding inierral of validity is larger than that given by the inequality (19). This special case often arises in practice. The inequality |/>o| < /2 is evidently satisfied, and the coefficients <r. depend upon «oj «i> •••> «»> ^n ••> ^» only: Co = «o> Ci = aibiy Ca = ttift, -h a,^, = «t*« "I + «.*?• Examples. 1) Cauchy gave a method for obtainhig the binomial theorem from the development of log(l + x). Setting y = ;xlog(l + x) = M(^j--+3-^-.-j. we may write (1 + x)** = eMioeO + ') = c» = l + ^+|^ + ---, whence, substituting the first expansion in the Mcond, /x x« x« \ u« /x «> x« \« If the right-hand side be arranged according to powers of c. It is ertdeni thai the coefficient of x" will be a polynomial of degree a in m whieh we ahaU eall P^(m). This polynomial must vanish when m = 0, 1» 2, • • •, a - 1, end moiS reduce to unity when fi = n. These facta completely detarmiBe P« la the foca <^^) ^"^ 1.2...» 2) Setting « = (1 + x)'/', where x lies between - 1 and -i- 1, we may writ* whore » = l.og(l + .)-l-| + f — • + (-1)--'^- 892 SPECIAL SERIES [IX, §183 The firet expansion is valid for all values of y, and the second is valid whenever |x|< 1. Hence the formula obtained by substituting the second expansion in the first holds for any value of x between - 1 and -f 1. The first two terms of this formuU are ,^)(l + x)i = .-|(l + l + ji^ + ... + :^^ + ...) + -- = e-?x + .... It follows that (1 + x)i/' approaches e through values less than e as x approaches Mro through positive values. 183. Division of power series. Let us first consider the reciprocal of a power series which begins with unity and which converges in the interval (— r, -h ?')• Setting y = bix -{- b^x^ -\ , we may write whence, substituting the first development in the second, we obtain an expansion for f(x) in power series, (25) f(x)=^l-b,x-^(bl-b,)x^ + .--, which holds inside a certain interval. In a similar manner a devel- opment may be obtained for the reciprocal of any power series whose constant term is different from zero. Let us now try to develop the quotient of two convergent power series 4*(^) _ gp + <^i^ + <^2^^ H Xl/(X) bo-\-biX+ b2X^-\----' If 6o is not zero, this quotient may be written in the form Then by the case just treated the left-hand side of this equation is the product of two convergent power series. Hence it may be written in the form of a power series which converges near the origin ; Clearing of fractions and equating the coefficients of like powers of 3P, we find the formulu3 DC. § \m] POWER 8£KI£8 S9S (27) a, = b,e, + *»«._, + . . . + i.e., !• - 0, 1, IV from which the coefficients ^» 0|» "•) 0. mij be wJ m Ut^ sively. It will be noticed that these ooefRoieoti Kn the those we should obtain by performing the diyision indioated by the ordinary rule for the division of polynomials arranged aoooidiiig to increasing powers of x. If ^0 = ^> th® result is different. Let us suppose for generality that \f/(or) = jr*^\J/i(x), where A; is a positive integer and ^,(jt) is a ])()wei* series whose constant term is not zero. Then we may write »(x) ^ 1 (^r,> and by the above we shall have also -p^ = Co + Ci* 4- • • • 4- e.^ix"'' + c»2J* + «t^,«**» 4- • • . It follows that the given quotient is expressible in the form (28) ii^ = $. + -£u + ... + £^. + ,.+ - -, , ^ ^ \f/(x) aj* a** ' X where the right-hand side is the sum of a ratio: .vhich becomes infinite f or x = and a power series whu .. ...... v. 5^^ near the origin. Note. In order to calculate the successive powers of a power Mrits, it is venient to proceed as follows. Assuming the identity (oo + aix + • • • + a^.x" + .••)"• = Co + Cii + ••• + c,x« + • • -, let us take the logarithmic derivative of each side and than d^r of frmrtions. This leads to the new identity 12Q\ (»»(ai + 2a«x+ • • + na,x--» + • • .)(eo + CiX + • • + c,x« + ♦• •) ^^' j = (ao + aix + . • + OnX* + • • '){ci + 2c«* + • • + «c.x— » + .••). The coefficients of the various powers of x are easily ealcalaied. Squat- ing coefficients of like powers, we find a aeqaMiM of fornmUs froai whidi Co, ci, . . ., c, . • • may be found saoce«lvdy if Co be known. It is evIdMl that 184. Development of l/Vl - Sxi + f«. Let us dtfelop l/VI - iM + f« according to powers of x. Setting y = 2xt - i«, we shaU have, when Iff < 1. y)-*=1 + ly+l^|f-^ . or (30) -_i=_ = i + ??!fi' + ?(l«-.V + -. 1 _., 1 vr -2x« + i« 894 SPECIAL SERIES [IX, §185 Collecting the terms which are divisible by the same power of «, we obtain an expansion of the form (81) ■ ^ = Po + Piz + Ptz^ + •" + P»«" + • • •, where Po = l, Pi = «, Pa = -^^» and where, in general, P» is a polynomial of the nth degree in x. These poly- nomials may be determined successively by means of a recurrent formula. Dif- ferentiating the equation (31) with respect to z, we find = Pi + 2P82 + • . . + nPnZ'*-! + • • •, (1 - 2X2 + 3fl)* or, by the equation (31), (X - 2)(Po + PiZ + • • • + P„2" + • • •) = (1 - 2x« + z^){Pi + 2P2Z + "). Equating the coefficients of z", we obtain the desired recurrent formula (n + l)P« + i = (2n + l)xP„ - nPn-i . This equation is identical with the relation between three consecutive Legendre polynomials (§ 88), and moreover Po = Xq, Pi = Xi, P2 = X2 . Hence P„ = X„ for all values of n, and the formula (31) may be written (82) , ^ = 1 + Xi 2; + X2 22 + . . . + x„ 2» + . . . , Vl - 2X2 + 2* where X, is the Legendre polynomial of the nth order X- = 5 — r(x2 - IH . 2.4.6...2n tte«'-^ '■■ We shall find later the interval in which this formula holds. II. POWER SERIES IN SEVERAL VARIABLES 185. General principles. The properties of power series of a single variable may be extended easily to power series in several independ- ent variables. Let us first consider a double series ^a„^x'"y'', where the integers m and n vary from zero to + 00 and where the coeffi- cients a^, may have either sign. If no element of this series exceeds a certain positive constant in ahsolute value for a set of values * = *o» y = yo> ^^« series converges absolutely for all values ofx and y which satisfy the inequalities |«| < |«o|, |y| < |2/o|- For, suppose that the inequality I««.a7y5|<3f or \a^,\< ^ l^oriyol" IX, $180] DOUBLE POWER SERIES |M is satisfied for all sets of values of m and fi. TbaD Uis abioloie valiM of the general element of the double series Sa.«z*y* is less than the corresponding element of the double series 1M\x/x^^\yfyJ^, Bat the latter series converges whenever |'|<|2^|> |y|<|yt|i and its sum is M ('-i^i)('-i^l) as we see by taking the sums of the elements by oolomns and adding these sums. Let r and p be two positive numbers for which the double ^la^nl'^p'* converges, and let R denote the rectangle formed by the four straight lines x = r, x = — Vf y = p, y =— p. For every point inside this rectangle or upon one of its sides no element of the double series (33) F(x,y) = 2a..a-y exceeds the corresponding element of the series Sjanil'^P* ^ abso- lute value. Hence the series (33) converges absolutely and uni- formly inside of R, and it therefore defines a continuous function of the two variables x and y inside that region. It may be shown, as for series in a single variable, ll»at the double series obtained by any number of term-by -term differen- tiations converges absolutely and uniformly inside the rectangle bounded by the lines a* = r — c, x = — r + €,y = p — t*, y»— ^ + €', where e and e' are any positive numbers less than r and ^ respec- tively. These series represent the various partial derivatives of F{x, y). For example, the sum of the series SfiMi.,«"-V is equal to cF/dx. For if the elements of the two series be arranged aeeofd* ing to increasing powers of a?, each element of the second series it equal to the derivative of the corresponding element of the ilrtt Likewise, the partial derivative d*-^'F/djrd^ is equal to th<» sum of a double series whose constant factor is a.,1.2-»' ^. Hence the coefficients e?^. are equal to the vmlues of the corrMpond- ing derivatives of the function F{x^ y) at the point x « y k 0, except for certain numerical factors, and the formula (SS) may be written in the form 896 SPECIAL SERIES [IX, §186 It follows, incidentally, that no function of two variables can have two distinct developments in power series. If the elements of the double series be collected according to their degrees in x and y, a simple series is obtained : (36) F(a;, y) = «^o + <^i + <A2 + • • • + ^« + • • • » where ^, is a homogeneous polynomial of the wth degree in x and y which may be written, symbolically, 1 / dF dFV'O The preceding development therefore coincides with that given by Taylor's series (§ 51). Let (a^, yo) ^ a Poi°* inside the rectangle R, and (xq + A, yo 4- k) be a neighboring point such that | iCo | + 1 ^ | < r, | yo | + 1 A; | < p. Then for any point inside the rectangle formed by the lines x = Xo±[r-\Xo\^y 3^ = yo±[p-|2/o|], the function F(Xf y) may be developed in a power series arranged according to positive powers of x — Xq and y — y^i (36) n^. + .,yo + ^) = Z ,;Sg..n ^'"^- For if each element of the double series be replaced by its development in powers of h and k, the new multi- ple series will converge absolutely under the hypotheses. Arrang- ing the elements of this new series according to powers of h and k, we obtain the formula (36). The reader will be able to show without difficulty that all the preceding arguments and theorems hold without essential altera- tion for power series in any number of variables whatever. 186. Dominant functions. Given a power series f(x, y^ z, -- •) in n yariables, we shall say that another series in n variables <^(ic, y,z,- •) damiruUes the first series if each coefficient of </>(a;, y, «, • • •) is positive and greater than the absolute value of the corresponding coefficient ^^ A*» Vt "t " •)• The argument in § 185 depends essentially upon IX. § iHB] DOUBLE POWER SSaiES SOT the use of a dominant function. For if the feriof S|«_,jr»y»| eoD* verges for x = r, y = p, the function where M is greater than any coefficient in the aeriee Sla^.r^W^L dominates the series 2a^,a!"y". The function ^{xy y) = is another dominant function. For the coefficient of jr*y' in \,{x, >/, is equal to the coefficient of the corresponding term in thr rxnan sion of Af(x/r -f y/p)'*-^*, and therefore it is at least equal to the coefficient of x^y* in <f>(x, y). Similarly, a triple series f{x, y, x) = Sa„^ary .I*, which converges absolutely for x ^ r^ y ^r'f M^r'*^ whei« r, r*, r" are three positive numbers, is dominated by an expreation of the form ^{Xy yyz) = ('-f)('-5)('-^) and also by any one of the expressions lff{x, y, z) contains no constant term, any one of the pr «<<*«itTy f Hp *ff sions diminished by M may be selected as a dominant fonetioo. The theorem regarding the substitution of one power sec i ei In another (§ 182) may be extended to power seriee in seTeral rariahlaa. If each of the variables in a amver^ertt pomer $miu in p v mriM m yiy ytf "t yp^^ replaced by a eo mver^m i jwwer ssrtts m f gart aU si xi, Xiy "jX^ which hcu no eomiant fsna, tks retuU tf Ms snAsTilii- tion may be written in the form of a power < to powers of x^, x^, ■, x^, provided iksU iks mU§t m t 4 mIm tf < of these variables is Uss thttn a 898 SPECIAL SERIES [IX, §186 Since the proof of the theorem is essentially the same for any number of variables, we shall restrict ourselves for definiteness to the following particular case. Let (37) i^(y,«)=2a„„2r«" be a power series which converges whenever | y | ^ r and | « | < r', and let ^ ' \ z = c^x -\- c^x^ -\ + c„a;«H be two series without constant terms both of which converge if the absolute value of x does not exceed p. If y and z in the series (37) be replaced by their developments from (38), the term in ?/'"«'• becomes a new power series in x, and the double series (37) becomes a triple series, each of whose coefficients may be calculated from the coeffi- cients a„„, h^f and c„ by means of additions and multiplications only. It remains to be shown that this triple series converges abso- lutely when the absolute value of x does not exceed a certain con- stant, from which it would then follow that the series could be arranged according to increasing powers of x. In the first place, the function /(y, z) is dominated by the function and both of the series (38) are dominated by an expression of the form (40) JL.-N=f^N($j, 1-- where M and N are two positive numbers. If y and z in the double series (39) be replaced by the function (40) and each of the products y**" be developed in powers of cc, each of the coefficients of the result- ing triple series will be positive and greater than the absolute value of the corresponding coefficient in the triple series found above. It will therefore be sufficient to show that this new triple series con- verges for sufficiently small positive values of x. Now the sum of the terms which arise from the expansion of any term ?/'»«'• of the •eries (39) is /x\^ + " ■^" VpI " (i-r"' IX, J 187] REAL ANALYTIC FfVrTTOVS 999 which is the general term of the seriea outamed by XDuitiplviog two series i(?)-Ui term by term, except for the constant factor J/. Both of (be series converge if x satisfies both of the inequalitiM ^ < pz-rn}' ' < P It follows that all the series considered will conrerge abtolatelj, and therefore that the original triple series may be arranged aooord- iug to positive powers of x^ whenever the absolute value of x U lets than the smaller of the two numbers pr/{r + N) and ^/{f + N), Note. The theorem remains valid when the series (38) oontmin constant terms b^ and Cq^ provided that |6o| "^^ ^ <u>d |<^|< r'. For the expansion (37) may be replaced by a series arranged according to powers oi y — bo and 2; — Co> by S 185, which reduces the disoos- sion to the case just treated. m. IMPLICIT FUNCTIONS ANALYTIC CURVES AND SURFACES 187. Implicit functions of a single variable. The existence of implieil functions has already been established (Chapter II, { 20et If.) luidw certain conditions regarding continuity. When the left-hand sides of the given equations are power series, more thorough investigmtioii is possible, as we shall proceed to show. Let F(Xy 1/)= be an equation wfune ^ft-kamd sids can be < in a convergent power series arranged aoeordmg to u of X — Xo and y — yo> v)here the eenstani term is sem cient of 1/ - y^f is different from Mere, Thm the equation ha$ erne mmd only one root which approaehee y^ae x approaekoe a^, emd tkmi rmi can be dei^eloped in a power series arramged artordinp to powers of X — x«. For simplicity let us suppose that x^^y^^O, which aaMmnts lo moving the origin of cottrdinates. Transposing the term of the first degree in y, we may write the given equation in the form (41) y =yi;x, y) = a,.z + a,oX* + <hi^ + ««/ + ' '. 400 SPECIAL SERIES [IX,§ia7 where the terms not written down are of degrees greater than the second. We shall first show that this equation can be formally sat- isfied bj replacing y by a series of the form (42) y = ciaj H- Cax' + . . • 4- c^x^ + • • • if the rules for operation on convergent series be applied to the series on the right. For, making.the substitution and comparing the coeffi- cients of Xj we find the equations and, in general, c, can be expressed in terms of the preceding c's and the coefficients a,^., where i + A; < n, by means of additions and multiplications only. Thus we may write (43) C, = PniP'lQl «20J ^\li ' -J «0»)> where P, is a polynomial each of whose coefficients is a positive integer. The validity of the operations performed will be estab- lished if we can show that the series (42) determined in this way converges for all sufficiently small values of x. We shall do this by means of a device which is frequently used. Its conception is due to Cauchy, and it is based essentially upon the idea of dominant functions. Let <^(x, y)=2i„„x'»y» be a function which dominates the function /(a;, y), where b^^ = J^j = and where b^^ is positive and at least equal to |a„,„|. Let us then consider the auxiliary equation (41') Y^^(^^^Y)=:S,b„„x^Y- and try to find a solution of this equation of the form (42^) Y=:C^x+ Caa;«-f ...-}-C,a;" + --«. The values of the coefficients Ci, C,, • • • can be determined as above, and are C, = *,o, C, = b,o + bnC, + bo,Cly ' . ., and in general It is evident from a comparison of the formulas (43) and (43') that |c,| < C,, since each of the coefficients of the polynomial P„ is positive and |a.J<6^,. Hence the series (42) surely converges DC, 5 187] REAL ANALYTIC FUNCTI0V8 401 whenever the series (42^) conyerges. Now we maj mImI tu th$ dominant fuii('fi«m ehi' r y\ tlie function where 3f, r, and p are three positive numbers. Then the auxiiuu^ equation (41') becomes, after clearing of fraction*, p + M p + M l__x^ r This equation has a root which vanishes for 2 = 0, namelj: 2(p4-Af) 2(p4-3/)\/^ p« l^l* The quantity under the radical may be written in the form (■-3(-9-' Hence the root }' may be written It follows that this root Y may be developed in a eeries which eon- verges in the interval (— a, -f <t), and this development must coin- cide with that which we should obtain by direct substitution, that is, with (42'). Accordingly the series (42) converges, a foriimrt^ in the interval (— or, + a). This is, however, merely a lower limit of the true interval of convcrutMU!** of tlie series <'42^. whir.h may be very much larger. It is evident from the mannrr :: determined that the sum of the series Mj, s.i:j,i,. :: . , , .a_ v ,U. Let us write the equation F(«, y) in the form y — /[«, y) « 0, and let y — P(x) be the root just found. Then if P(x) -f s be tabeti- tuted for i/ in F(7, y), and the result be arranged accordiDg lo powers of x and z^ each term must be divisible by «, sinre the whole expression vanishes when « = for any value of x. We aball ha^a then Fix, P(x) -f «] = xQ(x, x) , where Q(x, «) is a power teriee in m where 402 SPECIAL SERIES [IX, § 188 and z. Finally, if z be replaced by y — P{x) in Q(Xy z), we obtain the identity F{x,y) = ly-P(x)-]Q,{x,y), where the constant term of Qx must be unity, since the coefficient of y on the left-hand side is unity. Hence we may write (44) F{x, y) = [y - P{x)-](1 + ax + ^y + • • •)• This decomposition of F{xj y) into a product of two factors is due to Weierstrass. It exhibits the root y = P(x), and also shows that there is no other root of the equation F(xj y) = which vanishes with x, since the second factor does not approach zero with x and y. Note. The preceding method for determining the coefficients c„ is essentially the same as that given in § 46. But it is now evident that the series obtained by carrying on the process indefinitely is convergent. 188. The general theorem. Let us now consider a system of p equa- tions inp -\- q variables. 'Fi(a:i,X2, •..,«,; yi, y2, " •, yp)= 0, ^2(2:1, 0^2, •••,«,; yi,y2y"-,yp)=0, (46) Fp(xi,Xi,...,Xg] yi,2/2, •••,yp)=0, where each of the functions F^, F^, •■, Fp vanishes when Xi = y^ = 0, and is developable in power series near that point. We shall further suppose that the Jacobian D(Fi, F^, •••, Fp)/D{y^, 2/2, ••, ^p) does not vanish for the set of values considered. Under these conditions there exists one and only one system of solutions of the equations (45) of the form Vi = <^i(«i, a^a, '-,Xq), . . ., y^ = <^^, (^i, Xj, . . ., a;,), where <^i , <^a , • • , <^p are power series in x^, x^, • • ■, Xg which vanish when a?! = X, = . • . = iB, = 0. In order to simplify the notation, we shall restrict ourselves to the case of two equations between two dependent variables u and v and three independent variables x, y, and z : (46) )^i=aw +bv -{-ex -\-dy -\- ez +... = 0, F, = a'w -f Vv + c'x + c^'y 4. e'« ^ = 0. Since the determinant ab' - ba' is not zero, by hypothesis, the two equations (46) may be replaced by two equations of the form IX, $188] REAL ANALYTIC FUNCTIONS 408 (47) (u-Ja^^x-y-.'ii'-', where the left-hand sides contain no constant leruis and no tenu of the first degree in u and t;. It is easy to show, as iboTe, tfaH these equations may be satisfied formalLy by replacing m and v by power series in Xy y^ and z : (48) u = 2c,,,aryV, v = 2r;^,x'/«', where the coefficients e^,^, and el^, may be calculated from a^M^ and ^mnpqr ^J mcans of additions and multiplications only. In order to show that these series converge, we need merely compare them with tlie analogous expansions obtained by solving the two auxiliary equations — (.■^^;(..^) -'(--^> where M, r, and p are positive numbers whose meaning hat been explained above. These two auxiliary equations reduce to a single equation of the second degree 2p-f 43f^2p4-43/ ._x+x±f r which has a single root which vanishes for x = y s M £ e ^ 4(p + 23/) 4(p + 2Af) where a = r [p/(p + 43/)]^ This root may be developed in a convergent power series when- ever the absolute values of x, y, and « are all less than or equal to a/3. Hence the series (48) converges under the same oooditioos. Let wi and v^ be the solutions of (47) which are deralopable in series. If we set u = u^ -f m', v = rj + v' in (47) and artmnge tht result according to powers of x, y, «, «', v\ each of the terms mosl be divisible by u' or by v'. Hence, returning to the original ▼aria- bles Xj y, «, w, v, the given equations may be written in the form a7'^ Ut*-tiO/+(t'-«'i)^ =0, ^^'^ J(u-ti,)/i4-(p-«'i>^i = 0, 404 SPECIAL SERIES [IX, §189 where /, ^, /i , ^j are power series in a;, y, «, w, and v. In this form the solutions u=.ti^yV = Vi are exhibited. It is evident also that no other solutions of (47') exist which vanish for a; = y = « = 0. For any other set of solutions must cause /<^i — </>/i to vanish, and a comparison of (47) with (47') shows that the constant term is unity in both / and ^i, whereas the constant term is zero in both/j and ^; hence the condition /<^i — <j>fx = cannot be met by replacing u and v by functions which vanish when x = y = z=.0. 189. Lagntnge't formaU. Let us consider the equation (40) y = a + a^(y), where ^(y) is a function which is developable in a power series iny —a^ ^(y) = 0(a) + (y - o) <t>\o) + ^^^f^ r{a) + . . . , which converges whenever y — a does not exceed a certain number. By the general theorem of § 187, this equation has one and only one root which approaches a as z approaches zero, and this root is represented for sufficiently •mall values of x by a convergent power series y = a + aiz + a^x^ -\ . In general, if f{y) is a function which is developable according to positive powers of y — a, an expansion of /(y) according to powers of x may be obtained by replacing y by the development just found, (50) f{y) =fia) + ^ix + ^2X2 + . . . + ^^a.» + : . . ^ and this expansion holds for all values of x between certain limits. The purpose of Lagrange's formula is to determine the coefficients ■^Ij -0.2, • • •, Any ' • • in t«rms of a. It will be noticed that this problem does not differ essentially from the general problem. The coefficient A^ is equal to the nth derivative of AV) 'or y = 0, except for a constant factor n!, where y is defined by (49); and this derivative can be calculated by the usual rules. The calculation appears to be very complicated, but it may be substantially shortened by applying the fol- lowing remarks of Laplace (cf. Ex. 8, Chapter II). The partial derivatives of the function y defined by (49), with respect to the variables x and a, are given by the formuls [l-x«'to)]^ = «(„), tl-X«'(i,)]^ = l, we find Immediately <"' k"=*<<"- "f^^f K = /(y). On the other hand, It is easy to show that the formula IX,J18»] REAL ANALYTIC FUNCTIONS 4% b identically aatltfled, where F{y) is ad arbitimry fonelioa of y. f^ ■ide become! OD performing the indicated differentiations. We ahall now provw ^''^>[*^>'m for any value of n. It holds, by (61), for n = 1. In order to profo It la §m^ eral, let us assume that it holds for a certain number n. Then we dMll iHift But we also have, from (51) and (51'), whence the preceding formula reduces to the form dn^ which shows that the formula in question holds for all valaee of a. Now if we set X = 0, y reduces to a, u to /(a), and the nth derlfttito oi « with respect to 2 is given by the fonnula /a*u\ _ d— » \ax" /o ~ da"-» [^a)-/'(a)], Hence the development of /(y) by Taylor's series beooi (52) This is the noted formula due to LagraofB. It givet an 9t^ 9m\ fm for tlw root y which approaches zero as x i4)proaohes aero. Wo iriiaU tad lattr tiM limits between which this formula is appUetble. NoU. It follows from the general theorem that the root y, eo oiw oreda e a function of x and a, may be repreeented as a doublo Mvlee anrnofod a oc nwlit to powers of x and a. Thia aeriea can be obtaiaod Iff wplMlaC ••ch ol iko coefficients An by its development in power* of a. HoDOt the amim (M) moj be differentiated term by term with reepeet to o. Examples. 1) The equation (63) y = a + ?(»«-.l) 406 SPECIAL SERIES [IX, §190 hM one root wliich is equal to a when x = 0. Lagrange's formula gives the following development for that root : (W) xY d«-i (a2 - l)n 1 . 2 . . • n \2/ da^-i On the other hand, the equation (63) may be solved directly, and its roots are 1 . 1 « = - ± - VT^2ax + a;2. " X z The root which is equal to a when jc = is that given by taking the sign --. Differentiating both sides of (64) with respect to a, we obtain a formula which differs from the formula (32) of § 184 only in notation. 2) Kepler's equation for the eccentric anomaly m,* (66) u = a + e sin M , which occurs in Astronomy, has a root u which is equal to a f or e = 0. Lagrange's formula gives the development of this root near e = in the form e* d , . - , . , e" d^-Wsin^a) (66) u = a-h esma + — - — (sm^a) + • • • + — j— H + ' * •• 1.2 da 1.2. -n da»-i Laplace was the first to show, by a profound process of reasoning, that this aeries converges whenever e is less than the limit 0.66274^8 • • 190. Inversion. Let us consider a series of the form (67) y = 01X4-02x2 + .. • + a„a;» + ..-, where ai is different from zero and where the interval of convergence is(— r, + r). If y be taken as the independent variable and x be thought of as a function of y, by the general theorem of § 187 the equation (57) has one and only one root which approaches zero with y, and this root can be developed in a power series in y : (68) X = 6iy + biV^ + ftgy^ + • • • + ft^y" + . • • . The coefficients 6i, 62, 63 , • • • may be determined successively by replacing x in (67) by this expansion and then equating the coeflficients of like powers of y. The values thus found are a, a\ aj The value of the coefficient bn of the general term may be obtamed from Lagraoge's formula. For, setting ^(x) = ai + a2X + . • . + OnX^-'^ + • . ., tlM equation (67) may be written in the form ^(x) *Bm p. 918, Ex. 19; and Ziwkt, Xlements of TTieoretical Mechanict, 2d ed. p.aB6.— Teams. IX, 5 191] HEAL ANALYTIC FL.NLllU^b 407 and the development of the root of thto eqaatioo whleh appraiebM tmo villi 9 is given by Lagrange*! formula in the form %(0) where the subscript indicatea that we are to lei s b after indicated differentiations. The problem just treated haa 191. Analytic functions. In the future we shall say that a foii»> tion of any nuinber of variables x, y, s, ••• is analytic if it can be developed, for values of the variables near the point x^, f^, *%t "* in a power series arranged ai^cording to increasing powers of X — Xq, y — i/of « — «o> • • • which converges for sufficiently small values of the differences x — Xo, • • . The values which x,, y,, «,, • • may take on may be restricted by certain conditions, but we shall not go into the matter further here. The developments of the pree> ent chapter make clear that such functions are, so to speak, inter- related. Given one or more analytic functions, the operations of integration and differentiation, the algebraic operations of multipli> cation, division, substitution, etc., lead to new analytic functions. Likewise, the solution of equations whose left-hand member is ana- lytic leads to analytic functions. Since the very simplest functions, such as polynomials, the exponential function, the trigonometrie functions, etc., are analytic, it is easy to see why the first funetiocis studied by mathematicians were analytic. These functiona are still predominant in the theory of functions of a complex variable and ta the study of differential equations. Nevertheless, despite the fuodap mental importance of analytic functions, it must not be forgotten that they actually constitute merely a very particular group among the whole assemblage of continuous functions.* 192. Plane curves. Let us consider an arCi4B of a plane corva. We shall say that the curve is analytic along the art A B it the coordinates of any point M which lies in the neighborhood of any fixed point iUy of that arc can be developed in power series arranged according to powers of a parameter t — t^, ^ ^ ^y=^(0 = yo-f *i(<-^)+^(<--g« + -.+A.(f-f,)-+ . which converge for sufficiently small values of f — <,. • In the second volume an example of a n on a n a l yHc fteetlo« wHI heglvM, aUef whose derivatives exist thronghoot an Interval (a, 6). 408 SPECIAL SERIES [IX, §192 A point 3/0 will be called an ordinary point if in the neighbor- hood of that point one of the differences y — 2/0 ^ — ^q can be represented as a convergent power series in powers of the other. If, for example, y — Uo can be developed in a power series in (60) y-yo = cxip^ - x,)-\-<^%{^ -x,y+-" + c^{x - x,y + ..., for all values of x between Xf^ — h and x^ + h, the point {xq, y^) is an ordinary point It is easy to replace the equation (60) by two equations of the form (59), for we need only set iX = Xq-{- t — tQ, (^^> ?y = yo + Ci(^-A.)4---- + c„(^-g«+.... If Ci is different from zero, which is the case in general, the equa- tion (60) may be solved for a; — a^o in a power series in y — y^ which is valid whenever y — y^'is sufficiently small. In this case each of the differences x — Xq, y — Vq can be represented as a convergent power series in powers of the other. This ceases to be true if c^ is zero, that is to say, if the tangent to the curve is parallel to the x axis. In that case, as we shall see presently, cc — x^ may be devel- oped in a series arranged according to fractional powers of y — y^' It is evident also that at a point where the tangent is parallel to the y axis x — x^ can be developed in power series in y — y^^ but y — yo cannot be developed in power series ua x — x^. If the coordinates (a:, y) of a point on the curve are given by the equations (59) near a point Mq, that point is an ordinary point if at least one of the coefficients ai, b^ is different from zero.* If a-i is not zero, for example, the first equation can be solved for t — t^ in powers of x — a^o, and the second equation becomes an expansion of y — yo ^ powers oi x -^Xq when this solution is substituted for t - to. The appearance of a curve at an ordinary point is either the cus- tomary appearance or else that of a point of inflection. Any point which is not an ordinary point is called a singular point. If all the points of an arc of an analytic curve are ordinary points, the arc is said to be regular. • Thii condition Is iufficient, but not necessary. However, the equations of any curre, near an ordinary iM)lnt Mq, may always he written in sucli a way that ai and 6i do not both vanlnli, by a auitable choice of the parameter. For this is actually Moomplisbed in wiuatlous ((il). See also neeond footnote, p. 409. —Trans. IX, 5l»3] Kt.vL AJSALllIt; FUNCTIONS If each of the ooeffioients oi and 6| U lero, but a,, for exaiiipU, is different from zero, the first of equatioot (59) may be writleo is the form (x - Xo)* = (^ - Q^a^ 4- a, (<-<,)+. . .]*, whare the rifhi- liand member is developable according to powera ot t — t^. HeuoB t — Iq is developable in powers of (x — x,)*, and if < — /, in the second equation of (59) be replaced by that deyelopment, we obtain a development for y — y^ in powers of (x — a^)*: y - yo = <h(aJ - «•)+«!(« - a!,)'+ «•(« - aw)«+ ••. In this case the point (x^, y^ is usually a cusp of the fint kind.* The argument just given is general. If the derelopmeot of a; — Xo in powers of ^ — ^o begins with a term of degree ih y — |^ can be developed according to powers of (x — x^. The appearaooe of a curve given by the equation (69) near a point (ae^, y,) is of one of four types : a point with none of these peculiaritiea, a point of inflection, a cusp of the first kind, or a cusp of the aeoood kind.* 193. Skew curves. A skew curve is said to be analytic aUng an art AB if the coordinates x, y, z of b. variable point M can be dereloped in power series arranged according to powers of a parameter < — (« j-x = xo -I- ax(^ - <o) + ••• -H «.(< - <•)■ + •••» (62) \y = yo+f>,(t-to) + "+ b,(t - ^)' + ••., U = ;jJo + ci(^ - g -f- ... 4- c.(< -<,)•+ ..., in the neighborhood of any fixed point M^ of the arc. A point Mq is said to be au ordinary point if two of the three differences X — Xq, y — yof ^ — ^0 ^^^^ ^ developed in power series arranged according to powers of the third. It can be shown, as in the preceding paragraph, that the point Mq will surely be an ordinary point if not all three of the ooeiBeieota ai,bi, Ci vanish. Hence the value of the parameter t for a lingular point must satisfy the equations f dt ^' dt ^* dt ^' • For a cusp of the first kind the tangrat Um bttwwa Ik* two bcaaohM. Vbr a cusp of the second kind both branches lie on tho mmm aldo of tko laagoM. TW point is an ordinary point, of coane, U tho eoofloionti of tbo happen to be all zeros. —Trawi. t These conditions are not sniHeient to mako tho polat Jf«, whieh < a value (q of tho parameter, a stngalar point whoa a polat if of tho oarro aoar Ms corresponds to several values of t which approaeh !§ as if ayptoaohtt J^. 8eah Is the case, for example, at the origin oo tho cnnro il s fln s d hj tho oqaatkas « ■ ^. 410 SPECIAL SERIES [IX, §l&4 Let x^f yof «o be the coordinates of a point Mq on a skew curve T whose equations are given in the form (63) F(x, y, «) = 0, Fi (x, y, z) = 0, where the functions Fand F^ are power series m x — Xq, y — y^, z — z^. The point Mq will surely be an ordinary point if not all three of the functional determinants Z)(F, F,) ^ D(F, F,) ^ D(F, FQ I>(x, y) ' D{y, z) ' D(z, x) vanish simultaneously at the point aj = ccq, y = yo, z = Zq. For if the determinant D(F, Fi)/D(x, y), for example, does not vanish at 3/o, the equations (63) can be solved, by § 188, for x — Xq and y — y^ as power series in z — Zq. 194. Surfaces. A surface S will be said to be analytic throughout a certain region if the coordinates x, y, z of any variable point M can be expressed as double power series in terms of two variable parameters t — t^ and u — Uq «o = «io(^ — ^o) + aoi(^ — ""o) + )) + •••, ») + •••, in the neighborhood of any fixed point Mq of that region, where the three series converge for sufficiently small values of t —t^ and M — Mq. a point Mq of the surface will be said to be an ordinary point if one of the three differences x — Xq, y — y^j z — Zq can be expressed as a power series in terms of the other two. Every point Mq for which not all three of the determinants ^(y> ^) ^ D(z, x) ^ D(x, y) D{tj u)' D(t, u)' D(t, u) vanish simultaneously is surely an ordinary point. If, for exam- ple, the first of these determinants does not vanish, the last two of the equations (64) can be solved for t — tQ and u — Uq, and the first equation becomes an expansion of x — Xq in terms of y — yQ and « — «o upon replacing t — to and ti — Uq by these values. Let the surface S be given by means of an unsolved equation ^(*> y» *) = 0> and let Xq, y^, Zq be the coordinates of a point Mq of the surface. If the function F(x, y, z) is a power series in * ~ ^» y — yo» * — *o> and if not all three of the partial derivatives BF/di^f dF/dy^, dF/dzo vanish simultaneously, the point M^ is surely an ordinary point, by § 188. {x — Xq = aiQ(t — to) 4- aoi(^ — ""o) y - yo = *io(^ - ^o) + *oi {u - Wo) z- Zo = c^o(t - fo) + ^'oi (w - '^^o) lX,fl88j TRIGONOMETRIC SERIES 411 Note. The definition of an ordinary point on a cuire or 00 a ttir. face is independent of the choice of axes. For, let M^ (2^, y„ l^) be aa ordinary point on a surface S. Then the ooftrdinatee of any netgfa. boring point can"^ be written in the form (64), where not all three of the determinants D{y, z)/D(t, ti), Z)(«, x)/D{t, 11), Z)(x, y)/D{t, «) vanish simultaneously for t^t^.usxti^. Let us now aelect any new axes whatever and let A' = aia; + Ay 4-yi« + ^, K=a,x4-Ay + y,«4-«„ Z'!=ar,aj-|- Ay-|-y,« + S, be the transformation which carries x, y, x into the new codrdinatee A', }', Z, where the determinant A = /)(A', K, X)/D{x, y, «) is differ- ent from zero, lieplacing a-, y, z by their developments in series (64), we obtain three analogous developments for JT, K, iT ; and we cannot have D(t, u) D{ty u) D{t, u) for t=itQ, u = Uq^ since the transformation can be written in the form x = /l,A'-f Z?, r-f- C\Z -k-Dx, y = yi,A' + /<,r+r,z + ^, and the three functional determinants involving X, Y, Z eannoi vanish simultaneously unless the three involving ar, y, « alao Taniah simultaneously. IV. TRIGONOMETRIC SERIES MISCELLANF.ol'S SERIKS 195. Calculation of the coefficients. The series which we shall study in this section are entirely different from those studied above. Trigonometric series appear to have been first studied by D. Ber> noulli, in connection with the problem of the stretobed string. The process for determining the coeftl«i»*iiLs. which we are about to irivf. is due to Euler. I^t f{x) be a function defined in the interval (a, A). Wc shall first suppose that a and b have the values — w and + w, respec- tively, which is always allowable, since the subctitutioo .?•«»•— 412 SPECIAL SERIES [IX, §196 reduces any case to the preceding. Then if the equation (6S)Jlz) = ^-{'(aiC08x + bi8mx)-\ [•(a^cosmx-\-b„8mmx)-\ Z holds for all values of x between — ir and + tt, where the coefficients <*o» «i> ^i» •••>««> *iii> •• • are unknown constants, the following device enables us to determine those constants. We shall first write down for reference the following formulae, which were established above, for positive integral values of m and n : I sin mx rfx = ; I t087nxdx = {Sy if Wt>^0; I cos -mx cos wa; dx X"^ ' cos (w — w) X -f cos (m 4- n)x , ^ . - ^^ ^ — -z ^' ^— cfx = 0, 11 m ^ n; /"*"', , C^"" l-\-G0s2mx , .n ^ I co8*wia;tfa;= I ^ dx =7r, if m=?^0; I sin mx sin no; <fa; r cos (m — n)a; — cos Cm 4- n)x , ^ . „ = / ^^ ^ — 2 ^ — ^^— ^<fo; = 0, if m^Ti; X"*"' . a , r"*"' 1 — cos2ma5 , sm^'Twajdaj = j ^ (to =7r, if w =?t= Oj I sin mx cos Ttx cKx X sin (m + yi)a; + sin (m — n)x j 2 dx = 0. Integrating both sides of (65) between the limits - tt and + tt, the right-hand side being integrated term by term, we find (66) £'/(.)*.=|£'<^ which gives the value of a^. Performing the same operations upon the equation (66) after having multiplied both sides either by cos mx IX.} 190] TRIGONOMETRIC SERIES 418 or by sin mxj the only tenn on the right whoie integral and + TT is different from aero is the one in ooe'm« or in ein'eia. Hence we find the formulas / /(x) cosmxdx =i Tra., / f(x) mxdxwB vi^y respectively. The yalues of the ooefflcients may be etiiiinljlei] follows : (67) r 1 r^' I /* • 1 r*' b^ = — I /{a) sin ma da. The preceding calculation is merely formal, and therefore tent^ tive. For we have assumed that the function /(i) can be developed in the form (65), and that that development oonrergee uniformly between the limits — w and + tt. Sinoe there is nothing to prove, a priori, that these assumptions are justifiable, it is essential that we investigate whether the series thus obtained converges or noL Replacing the coefHcients a^ and bt by their values from (67) and simplifying, the sum of the first (m + 1) terms is seen to be h f /wri+co8(^-*)+«^2(*-*)+-+***"»(*~')i^- But by a well-known trigonometric formula we have . 2»+l - -h cos a -h cos 2a H h cos smi*« " — » 2sin| whence • 2m-|-l, . 1 r*' "° — 2~^''"*) ^^- 2sin^^ or, setting a = a; -f 2y, (68) 5... = -J _/[« + 2y) \i„y '' <'»■ t The whole question is reduced to that of finding the limit of Ihit siun as the integer m increases indefinitely. In order to stodj this question, we shall assume that the function /[x) salis6et «!• fol- lowing conditions : 414 SPECIAL SERIES [IX, §196 1) The function f(x) shall be in general continuous between — ir and -f TT, except for a finite number of values of x, for which its value may change suddenly in the following manner. Let c be a number between — tt and -|- tt. For any value of c a number h can be found such that fix) is continuous between c — h and c and also between c and c-\- h. As c approaches zero, f{c + c) approaches a limit which we shall call /(c -f 0). Likewise, f(c — e) approaches a limit which we shall call f(c — 0) as c approaches zero. If the fimction f(x) is continuous for a; = c, we shall have /(c) =/(c + 0)=/(c — 0). If f(e -f 0) ^ f{c — 0)jf(x) is discontinuous for x = c, and we shall agree to take the arithmetic mean of these values [/(c + 0) +/(c — 0)]/2 for /(c). It is evident that this definition of /(c) holds also at points where f(x) is continuous. We shall further suppose that /(— tt + c) and /{it — c) approach limits, which we shall call /(— tt + 0) and /(tt — 0), respectively, as c approaches zero through positive values. The curve whose equation is y =f{^) must be similar to that of Fig. 11 on page 160, if there are any discontinuities. We have already seen that the function f(x) is integrable in the interval from — TT to 4- TT, and it is evident that the same is true for the product oif(x) by any function which is continuous in the same interval. 2) It shall be possible to divide the interval (— tt, + tt) into a finite number of subintervals in such a way that/(£c) is a monoton- ically increasing or a monotonically decreasing function in each of the subintervals. For brevity we shall say that the function f{x) satisfies Dirichlefs conditions in the interval (— tt, + tt). It is clear that a function which is continuous in the interval (— tt, + tt) and which has a finite number of maxima and minima in that interval, satisfies Dirichlet^s conditions. 196. The integral J^ f (x) [sin nx/sin x] dx. The expression obtained for 5^^, leads us to seek the limit of the definite integral X' sin nx . /(«) — dx fo sm a; 88 n becomes infinite. The first rigorous discussion of this ques- tion was given by Lejeune-Dirichlet.* The method which we shall employ is essentially the same as that given by Bonnet. f • CfrttU's Journal, Vol. IV, 1829. t MHnoiru det tavanta strangers publics par I'Acaddmie de Belgique, Vol. XXIIL DC, $198] TRIGONOMETRIC SERHES 416 Let us first consider the integral (69) y=jrV)?i^^. where A is a positive number less than w, and 4(x) U a which satisfies Dirichlet's conditions in the interval (0, A). If 4(«) is a constant C, it is easy to find the limit of J, For, iiitlim y ■* <»b we may write '-X T*"' and the limit of / as n becomes infinite is Cw/2, by (99), f 176. Next suppose that ^(2^) is a positive monotonically deerMtfaif function in the interval (0, h). The integrand ohanget tigii for all values of x of the form kw/n. Henoe J may be written where tti. = (* + !)» sinn^ X " "^^'^ dx and where the upper limit A is suppoeed to lie between wtw/m and (7n -f- l)7r/7i. Each of the integrals u^ is less than the preoeding. For, if we set na; = A*7r 4- y in u^, we find JC'^ / y-h Anr \ siny . and it ia evident, by the hypotheses regarding ^('), that this inl»> gral decreases as the subscript k increases. Henoe we shall haT« the ecjuations / = tto - (t«i - V,) - (m, - k«) , y = t£„ - w, -I- (14 - u.) 4- (|^ - «») + • • • . which show that J lies between «, and ^ — Ut, It follows thai J it a positive number less than u,, that is to 9Mj, lest than the intflfiml X ^ , . sm IMP . -^ But this integral is itself less than the integral where .1 denotes the value of the definite integral ^*[(«n fi/y]^ 416 SPECIAL SERIES [IX,§19« The same argument shows that the definite integral r* . , ^ sin na; , where c is any positive number less than h, approaches zero as n becomes infinite. If c lies between (t — l)'7r/n and iw/n, it can be shown as above that the absolute value of J^ is less than smnx <f>{x) dx and hence, a fortiori, less than <^(c) (ijr _\. \n} IT ^ 27r <^(c) c Vw / iir n n c Hence the integral approaches zero as n becomes infinite.* This method gives us no information if c = 0. In order to dis- cover the limit of the integral /, let c be a number between and hj such that <^(a;) is continuous from to c, and let us set ^{x) = ^(c) 4- ^{x). Then xj/ix) is positive and decreases in the interval (0, c) from the value <^(+ 0) — <^(c) when a; = to the value zero when a; = c. If we write / in the form =*(c)jr'5i sm nx dx -h X\ . . sin Tw; - , ^(aj) — :;; — dx -\- and then subtract (7r/2)<^(+ 0), we find X *(^) - sinTiaj dx (70) Ja ^ Jc sinn«c In order to prove that J approaches the limit (ir/2) <^(-f- 0), it will be sufficient to show that a number m exists such that the absolute • ThU result m»y be obtained even more simply by the use of the second theorem of the mean for integrals (§75). Since the function ^(x) is a decreasing function, that formula gives ABd Um right-hand member evidently approaches zero. IX, $196] TRIGONOMETRIC SERIES 417 value of each of the terms on the right U leM than a pMMtifBfld positive number (/4 when n is greater than m. By the remark made above, the absolute value of the integral /■ is less than A ^(-}- 0) = i4 [^(+ 0) — ^(f)]. Since ^{x) approachm <^(+ 0) as X approaches zero, e may be taken so near to zero thai A [<^(+ 0) - <^(c)] and (•»r/2)[^(+ 0) - ^c)] are both less than t/4. The number c having been chosen in this way, the other two terma on the right-hand side of equation (70) both approaeh laro as a becomes infinite. Hence n may be chosen so large that the abeo- lute value of either of them is less than «/4. It followt that (71) limy = f^+0). We shall now proceed to remove the various restrictions which have been placed upon <f>{x) in the preceding argument. If 4(') >* a monotonieally decreasing function, but is not always poattive, the function ^(a;) = <^(a;) + C is a positive monotonieally deereaatng fime- tion from to A if the constant C be suitably chosen. Then the formula (71) applies to ^(^). Moreover we may write Jr\, ^ sinnar , f*. aintur, ^ f * sin nx ^ and the right-hand side approaches the limit (w/2)^+0) — (w/2)Cp that is, (7r/2)<^(+0). If <f>(x) is a monotonieally increasing function from to A, — 4(ar) is a monotonieally decreasing function, and we shall have XV)^^=-X'-*^'^ -Jx. Hence the integral approat^hes (^/2)o Finally, suppose that ff>{x) is any fu: - > — Dirich- let's conditions in the interval (0, A). Then the interval (0, k) may be divided into a finite numbt^r of subintervaU (0, «), (a, k), (P> c\ • • •, (^» A)' in ^^^ of ^'^^'^^* ^(') ** * monotonieally increasing or decreasing function. The integral from to a approaches the limit (7r/2)<^(4- 0). Each of the other integrals, which are of the type H^f\( ^ Sin nx , x)——dx, 418 SPECIAL SERIES [IX, §197 approaches zero. For if <l>(x) is a monotonically increasing function, for instance, from a to 6, an auxiliary function ^(ic) can be formed in an infinite variety of ways, which increases monotonically from to ft, is continuous from to a, and coincides with <f}(x) from a to b. Then each of the integrals X" , ^ sin no; , r^ sinnx approaches ^(-f- 0) as w becomes infinite. Hence their difference, which is precisely H, approaches zero. It follows that the formula (71) holds for any function <t>{x) which satisfies Dirichlet's condi- tions in the interval (0, h). Let us now consider the integral (72) ^=X/(-)l^'^' 0<A<,r, where f(x) is a positive monotonically increasing function from to A. This integral may be written -X'[/<-)ii^]^*- and the function <l>(x) = /(x) ic/sin a; is a positive monotonically increasing function from to h. Since /(+ 0) = <^(+ 0), it follows that (73) lim/ = |/(+0). This formula therefore holds if f(x) is a positive monotonically increasing function from to h. It can be shown by successive steps, as above, that the restrictions upon f(x) can all be removed, and that the formula holds for any function f(x) which satisfies Dirichlet's conditions in the interval (0, h). 197. Fourier series. A trigonometric series whose coefficients are given by the formulae (67) is usually called a FouHer series. Indeed it was Fourier who first stated the theorem that any function arbi- trarily defined in an interval of length 27r may be represented by a series of that type. By an arbitrary function Fourier understood a function which could be represented graphically by several cur- vilinear arcs of curves which are usually regarded as distinct curves. We shall render this rather vague notion precise by restricting our discussion to functions which satisfy Dirichlet's conditions. IX.§1!)7] TRIGONOMETRIC 8BRIE8 419 In order to show that a function of this kind can be Mipw Hwu liMi by a Fourier series in the interval {— w, -^ w), we mttst find t^ limit of the integral (68) as m becomes infinite. Lei us divide this integral into two integrals whose limite of integrataon mm and (IT - a-)/2, and - (ir + x)/'J and 0, respeetively, and let nt make the substitution y =a — z in the seoood of theee tntagrml^ Then the formula (68) becomes frJo '^^ ' sm« When X lies between — ir and + w, (w — x)/2 and (w -♦- ar)/2 both lie l)etween and tt. Hence by the last article the right-hand side of the preceding formula approaches 1 [E^(. + 0) + lA^ - 0)] = 2L' >o)±2][£^ as m becomes infinite. It follows that the series (65) oooTerges and that its sum is /(a;) for every value of x between — w and + ». Let us now suppose that x is equal to one of the limits of the interval, — ir for example. Then ^'.^.j may be written in the fom The first integral on the right approaches the limit/t—ir + •)/!. Setting y = TT — « in the second integral, it takes the form ^Jo • Z /(^ ,. gin(2m.H)s^^ sins whicli approaches /(7r - O)/!'. Hence the sum oi ihe ir.. series is [./(tt - 0) -\- f{- w + 0)]/2 when ar = - w. It . • • that the sum of the series is the same when » «+ w. If, instead of laying off x as a length along n strmight iinr, ww lay it off as the length of an arc of a unit circle, connling in Um 420 SPECIAL SERIES [IX, §197 positive direction from the point of intersection of the circle with the positive direction of some initial diameter, the sum of the series at any point whatever will be the arithmetic mean of the two limits approached by the sum of the series as each of the variable points m' and m", taken on the circumference on opposite sides of m, approaches m. If the two limits /(— tt + 0) and /(tt — 0) are different, the point of the circumference on the negative direction of the initial line will be a point of discontinuity. In conclusion, every function which is defined in the interval (— TT, 4- tt) and which satisfies Dirichlefs conditions in that inter- vcU may be represented by a Fourier series in the same interval. More generally, let f(x) be a function which is defined in an interval (a, a -f 27r) of length 27r, and which satisfies Dirichlet's conditions in that interval. It is evident that there exists one and only one function F(x) which has the period 27r and coincides with f(x) in the interval (a, a + 27r). This function is represented, for all values of x, by the sum of a trigonometric series whose coeffi- cients a^ and b^ are given by the formulae (67): 1 r^" 1 r^"" dm — - j F{^) cos mxdx, b„= - j F(x) sin mx dx. The coefficient a^, for example, may be written in the form am = ~ I ^M COS mxdx-{-- I F(x) cos mx dx , where a is supposed to lie between 2A7r — tt and 2A7r + tt. Since F(x) has the period 27r and coincides with f(x) in the interval (a, a 4- 27r), this value may be rewritten in the form (74) <*« = -/ f{x) COS m^dx-\- \ f{x) cos mx dx = — I f{x) COS mx dx. ^ c/ar Similarly, we should find -« /»<r-flir (75) h^=i- i f{x)%mmxdx. Whenever a function /(x) is defined in any interval of length 27r, the preceding formulae enable us to calculate the coefficients of its development in a Fourier series without reducing the given interval to the interval (- tt, -f tt). IZ.flW] TRIGONOMETRIC SERIES 421 191. ExampUt. 1) Let tu And a Foaiter ttrlti wboat mm to - 1 i « < z < 0, and +1 for < x < + «. TU ionBBtai (67) glf* ih* vsIims 1 /•• 1 /»» 00 = - / -<te + - I dacsO, \. = i r*-«inmxdx + i /*'■«.. -..^ - > - «»"*^ - «»(- »^ If m is eyen, b^ is zero. If m la odd, 6. is 4/m«. MulUplylnf all tba cients by ^/4, we see that the sum of the (76) Binx sinSx rin(«m-H)« + 1 is - 9r/4 for - )r < z < 0, and + fc/A for < x < r. Tba pofaH « k Ohm polst of discontinuity, and the sum of the leriea is zero when x = 0, as it rtpmM ba. More generally the sum of the series (76) is «/4 when slnx ia poilU?*, r'i when sin x is negative, and zero when sin z = 0. The curve represented by the equation (76) ia oompoaed of an tnflBlla namber of segments of length n of the straight lima y = ± r/4 and an Infialla ^'^m- ber of isolated points (y = 0, z = kit) on the z axis. 2) The coefficients of the Fourier derelopment of z in the intertai from 10 2n are ao= - / zdz = 2*, 1C Jo 1 f" , rz sin mz-l«» . 1 /•••. . ^ a^=-l xcosmzdz= |___ J^ + _ j^ *im.*.0. . 1 r*' 1 ^ rzcoamx-l«» . 1 /•»» . S 6,n=-/ zsinmzdz = - — — — — + — - / ooaiwAa Hence the formula m\ z _ y ainx sinax atnSx ^^ 2"2 1 2 ~ S is valid for all values of z between and Sir. If wa aM y aqoal lo tka avfM tm the right, the resulting equation repreanta a eorraooapoiaddf ■• tetidlaBaa- ber of segments of straight lines parallel to y = s/1 and aft laSnlto MUBbw of isolated points. NoU. If the function f{x) deflnad in tha intarral (- v, -t- a) Is mm, IkM la to say, if /(- z) =/(z), each of tte aoaOolaBta tb to aaro, aiaoa it to ainaixdii. J*/l(z)ainmxdx = -^/[x) Similarly, if /(z) is an odd fanettOB, that to. If /[ - x) = -/tx). earh of the coefficients a« i^ zero. Including d*. A fooetioB /fx) whieh to daOned oolj tn 422 SPECIAL SERIES [IX, §199 the interval from to »r may be defined in the interval from - tt to by either of the equations , . ^, . f{-x)=f{x) or /(-x) = -/(x) If we choose to do so. Hence the given function /(x) may be represented either by a series of cosines or by a series of sines, in the interval from to ;r. 199. Expansion of a continuous function. Weierstrass' theorem. Let /(x) be a function which is defined and continuous in the interval (a, 6). The following remarkable theorem was discovered by Weierstrass : Given any preassigned posi- tive number «, a polynomial P(x) can always be found such that the difference /(x) _ p(x) is less than c in absolute value for all values of x in the interval (a, b). Among the many proofs of this theorem, that due to Lebesgue is one of the simplest* Let us first consider a special function \p{x) which is continuous in the interval (- 1, + 1) and which is defined as follows : \p{x) = for - 1 <x < 0, f (x) = 2ifcx for < X < 1, where A; is a given constant. Then yf/{x) = (x + | x |) A;. Moreover for - 1 < x < + 1 we shall have |xi = vr^(i-x2), and for the same values of x the radical can be developed in a uniformly con- vergent series arranged according to powers of (1 - x^). It follows that |x|, and hence also ^(x), may be represented to any desired degree of approximation in tne interval (- 1, + 1) by a polynomial. Let us now consider any function whatever, /(x), which is continuous in the interval (a, 6), and let us divide that interval into a suite of subintervals (oo, ai), (ai , az), • • • , (a„-i , a„), where a = ao < ai < az < • • • < On-i < a„ = 6, in such a way that the oscillation of f{x) in any one of the subintervals is less than c/2. Let L be the broken line formed by connecting the points of the curve y = /(x) whose abscissae are ao , ai , ag , • • • , &. The ordinate of any point on L is evidently a continuous function 0(x), and the difference f{x) — <f>{x) is less than e/2 in absolute value. For in the interval (a^a-i, a^), for example, we shall have fix) - 0(x) = [fix) -/(a^-i)] (1 - ^) + [fix) -fia^)]0, where x — 0/4_i = tf(a,i — o^_i). Since the factor d is positive and less than unity, the absolute value of the difference f — <f> is less than c(l — ^ + 0)/2 = c/2. The function 0(x) can be split up into a sum of n functions of the same type as f (x). For, let ^0, -4i, ^2, • • • , ^n be the successive vertices of L. Then 0(x) is equal to the continuous function ^i (x) which is represented throughout the interval (o, 6) by the straight line AqAi extended, plus a function 0i(x) which is represented by a broken line AoAi- •- Ai, whose first side AqAi lies on the X axis and whose other sides are readily constructed from the sides of L. Again, the function <pi (x) is equal to the sum of two functions ^2 and 02 » where ^2 is zero between Oo and ai, and is represented by the straight line A1A2 extended between ai and 6, while 02 is represented by a broken line AoAi'Ai- • • An whose first three vertices lie on the x axis. Finally, we shall obtain the equation ^ = ^i + ^1 + • • • + ^H , where ^< is a continuous function which vanishes between Oq and ot-i and which is represented by a segment of a straight line * Bulletin dea sciences matMrnatiquen, p. 278, 1898. lX,§aOO] TRIGONOMETRIC SKRIKS 42| between o^.i and 6. it we tuen maxe uie suimiuiuuuii X b «k 4. i^ wImto m and 71 aru suitably choten numben, the fonsikMi f #(») Btj bt ditetd Is Ite interval (- 1, -f 1) by tbe equation f,(x) = t<Jr + |X|). and hence it can be repreaented by a polynomial with any rtMJml dtfiM oC approximation. Since each of the functiona f<(z) can be repiMMiad la Ik* interval (a, 6) by a polynomial with an error le« than i/Sa, It la sum of these polyuomiala will differ from/(x) by leea than «. It foUowH from the preceding theorem that any /untUom /{x) wkkk <■ nous in an interval (a, 6) inay be rtprtteniti by an ii^niU mrim ^ jrtrfrnirmiah Mohich converges uniformly in that interval. For, let <i , c«, .,<.,.•. bt a 1 of positive numbers, each of which is less than the preceding, where ^1 zero aa n becomes infinite. By the preceding theorem, corraqModlBf to caieii of the c*8 a polynomial P, (x) can be foond anch that the dUlereoee /(«) ~ Pi{g)lt leas than e, in absolute value throughout the interval (a, 6). Then tbe amim Pi(x) + [P,(x) - P,(x)] + .. . + [P.(x) - P.-i(*)) + ••• converges, and its Hum is/(x) for any value of x iiiaide the interval (a, 6). For the sum uf the tirst n terms is equal to P.fx), and the differeooe /(x) — d^, whleh is less than Cn , approaches zero as n becomes infinite. Moreover th* ■artea cem- verges uniformly, since the absolute value of the difference /(x) — 8m win be Icai than any preaBsi<,'ned positive number for all valuea of n which eieead m eeftnln fixed integer N, when x has any value whatever between a and 6. 200. A continuous function without a derivative. We ahall ooodudo thii ebaplor by giving an example due to Weierstraas of a oontlnoona fnnctioo whlell does not possess a derivative for any value of the variable whatever. Let 6 bt a poil- tive constant less than unity and let a be an odd Inttfer. Then the ftnwrtf F{x) defined by the convergent infinite aeries (78) F(x) ='^b^coa{c^ xz) is continuous for all values of x, ainct the teriea eoowftt unUotmif hi aaj interval whatever. If the product ab la lea than unity, tht hold for the series obuined by term-by-term dlflettB th l tfc wi, tion F{z) possesses a derivative which ia ittelf a eontinnoat fkuelkm. Wt Anil now show that the sUte of affairs is eteentially difftrtnt II tht a certain limit. In the first place, setting M-l S« = i V fr-:oos[a-ir(x + A)) - coe(«««x)}, ■»■•» ii, = 1 2^ 6- {cot [a- r(« + A)l - oot (•• w)) . we may write 424 SPECIAL SERIES [IX, §200 On the other hand, it is easy to show, by applying the law of the mean to the function coe(a"«'x), that the difference co8[a'»«'(x + h)] — coa {a"* tcx) is less than iro" \h\in absolute value. Hence the absolute value of Sm is less than m-l X a"» o»» — 1 a**©" = Tf — - — — - > ao — 1 and consequently also less than «r(a6)'»/(a6 — 1), if oft >1. Let us try to find a lower limit of the absolute value of R„ when h is assigned a particular value. We shall always have a'»x = am + lm, where a>, is an integer and fm Hes between — 1/2 and + 1/2. If we set 1. ^m Cm a™ where c is equal to ± 1, it is evident that the sign of h is the same aa that of C, and that the absolute value of h is less than 3/2a"'. Having chosen h in this way, we shall have a" ir{x ■\- h)z= a^-f^a"^ 7t{x + h) = a^-"^7t{am + fim) • Since a is odd and e™ = ±1, the product a»-*»{am + em) is even or odd with a^ -f- 1, and hence cos[a»;r(x + A)] = (- l)«m+i. Moreover we shall have C08(a»;rx) = cos(a«-»«a'»;rx) = cos[a»-"»;r(aTO + ^m)] = cos (a"-*" OTm ?r) cos (a»-»* ^^ tt) , or, since a"-"»a« is even or odd with a^ cos{a^jtx) = (-l)«i»cos(a'»-"»fm^). It follows that we may write Rn.= (_!)«« + ! + 00 2)&"[l+cos(a»-'»^,„;r)]. Since every term of the series is positive, its sum is greater than the first term, and oonaeqaently it is greater than 6« since U lies between - 1/2 and + 1/2. Hence or, since |A|<3/2a"', If a and h satisfy the inequality (80) a6>l + ^, we shall have 2 / rv„ ^ 7r(a6)» whence, by (79), IX. Exs] EXERCI8E8 4S5 As m becomes infinite the expre«ion on tiM tttreae rifbt : while the absolute valtte of h ftpproaehee mto. rninigpillj, -w small < be choeen, an increment h can be found which to lea ihaa < la lute value, and for which the absolute value of [F{z + A) - F{M)]/k aaaate aay preaesigned number whatever. It follows thai if a and 6 aitiiCy the nlitlna flO), the function F{x) possesses no derivative for any value of « ixnciSM 1. Apply Lagrange^s formula to derive a develofmiat tai powws of s df Iteft root of the equation y< = ay + z which to equal io a wbao • a 0. 2. Solve the similar problem for the equation f ^ a-^ t^*^ 8 0. Apply tkt result to the quadratic equation a - te + cz* = 0. D«vtlop la powwi of e tkaft root of the quadratic which approaches a/6 as c approoehss asro. 3. Derive the formula 4. Show that the formula X VI holds whenever z is greater than - 1/2 6. Show that the equation 1 2z x = 2.4V1 + W ^2.4.«Vl + W 2 l + «« holds for values of z less than 1 in absolute value. What to the sum of the when 1 z I > 1 ? 1 r nx ^ njn-l) / x V it(«-l)(ii>t) / « V 1 a-L a + x"^ 1 2 \o + xy"" l.a.S \«+«/ J 6. Derive the formula (a + z)- 7. Show that the branches of the funetloB iiBflisaBd« to and 1, respectively, when sinx = are do fl opa b to i& powers of sin X : 1.2 1 . » . o . S [Make use of the differential equation which is satisfied by u = oosmx and by « s iin ms, whors y v ilncj 8. From the preceding fonnotos dodoos cos (n arc COS x), da(«af«oai«). CHAPTER X PLANE CURVES The curves and surfaces treated in Analytic Geometry, properly speaking, are analytic cui'ves and surfaces. However, the geomet- rical concepts which we are about to consider involve only the exist- ence of a certain number of successive derivatives. Thus the curve whose equation is y =f(x) possesses a tangent if the function f(x) has a derivative f'(x) ; it has a radius of curvature if f'(x) has a derivative f"(x) ; and so forth. I. ENVELOPES 201. Determination of envelopes. Given a plane curve C whose equation (1) • f(x, y,a) = involves an arbitrary parameter a, the form and the position of the curve will vary with a. If each of the positions of the curve C is tangent to a fixed curve E, the curve E is called the envelope of the curves C, and the curves C are said to be enveloped by E. The problem before us is to establish the existence (or non-existence) of an envelope for a given family of curves C, and to determine that envelope when it does exist. Assuming that an envelope E exists, let (x, y) be the point of tan- gency of E with that one of the curves C which corresponds to a cer- tain value a of the parameter. The quantities x and y are unknown functions of the parameter a which satisfy the equation (1). In order to determine these functions, let us express the fact that the tangents to the two curves E and C coincide for all values of a. Let hx and By be two quantities proportional to the direction cosines of the tangent to the curve C, and let dx/da and dy/da be the derivatives of the unknown functions x = </>(«), y = \j/{d). Then a neoessary condition for tangency is dx dy 426 X,jaoiJ ENVELOl'Ks 4IJ On the other hand, since a in equAtion (1) haa % eoutont Taloa for the particular curre C considered, we shall have which determines the tangent to C. Again, the two unknown fua^ tions X = ^(a), y = ^(a) satisfy the equatioo yi*»y, «) = 0, also, where a is now the independent variable. Uenot or, combiniDg the equations (2), (3), and (4), (5) lf-0. The unknown functions x — ^(a), y = ^a) are aolatioof of thia eqa^ tion and the equation (1). Hence the equation of the enp^pe^ im ease an envelope exists^ is to be found by eliminatiny the partm^Ur^ between the equations / = 0, df/ca = 0. Let R{x, ^) = be the equation obtained hf eliminating a b e t wee a (1) and (5), and let us try to determine whether or not this equatioQ represents an envelope of the given curves. Let C, be the pariien* lar curve which corresponds to a value a^ of the parameter, and lei (^0) Vo) ^ t^6 coordinates of the point A/, of intemettOD of Um two curves (6) /(x,y,«o) = 0, ^ = 0. The equations (1) and (5) have, in general, Bolutiona of the form X = <^(a), y = ^(a), which reduce to z^ and y«, retpMltTelj, for a = Oy. Hence for a = a^ we shall have dxAda)^^ dyAda). This equation taken in connection with the equation (3) ahovt that the tangent to the curve r, coincides with the tangeoi to tlie curve described by the point (x, y), at least unless df/dx and df/^ are both zero, that is, unless the point -V, is a singular point for the curve Co. It follows that the equation /?(x, y) - reprmrmiM either the envelope of the eurvei C or eUe the hens ^ iinpUmr points om these curves. 428 PLANE CURVES [X,§202 1 his result may be supplemented. If each of the curves C has one or more singular points, the locus of such points is surely a part of the curve /?(a;, y) — 0. Suppose, for example, that the point (x, y) is such a singular point. Then x and y are functions of a which satisfy the three equations f{x.y,a) = 0, ^ = 0, g = 0, and hence also the equation df/da = 0. It follows that x and y satisfy the equation R{Xj y) = obtained by eliminating a between the two equations / = and df/da = 0. In the general case the curve R(Xf y) = is composed of two analytically distinct parts, one of which is the true envelope, while the other is the locus of the singular points. Example. Let us consider the family of curves f{x,y,a) = y'-y' + {x-ay = 0. The elimination of a between this equation and the derived equation |f = -2(x-«) = gives y* — y* = 0, which represents the three straight lines y = 0, y = + l, y= — 1. The given family of curves may be generated by a translation of the curve y* — y^ + a;^ = along the x axis. This curve has a double point at the origin, and it is tangent to each of the straight lines y = ± 1 at the points where it cuts the y axis. Hence the straight line y = is the locus of double points, whereas the two straight lines y = ± 1 constitute the real envelope. 202. If the curves C have an envelope E, any point of the envelope is tht limiting position of the point of intersection of two curves of the family for which the valvss of the parameter differ by an infini- tesimal. For, let (7) f(x, y, a) = 0, f(x, y, a -{- h) = be the equations of two neighboring curves of the family. The equations (7), which determine the points of intersection of the two coryes, may evidently be replaced by the equivalent system X, § 'M-2] ENVELOPES 4f9 the seoend of which reduces to d//da » m A approtehei »ro, ihal is, as the second of the two curves approaches the first, Thi« pto^ erty is fairly evident geometrically. In Fig. 37, a, for injunca, thm point of intersection ^V of the two neighboring curves C and C* approaches the point of tangencj M na C ipproaohef Um enrrs C Fia. 37, a FW. 37, 6 as its limiting positica. Likewise, in Fig. 37, &, where the pi von curves (1) are supposed to have double points, the point of intersoc- tion of two neighboring curves C and C ^>proaehes the point wber* C cuts the envelope as C approaches C. The remark just made explains why the locus of singular potots is found along with the envelope. For, suppose that J\x^ y, a) is a polynomial of degree m in a. For any point A/«(aE;t, y«) eboaaa at random in the plane the equation (8) • Ax^,y.,a)^0 will have, in general, m distinct roots. Through such a point theiv pass, in general, m different curves of the given family. But if tha point Mq lies on the curve /?(x, y) = 0, the equations are satisfied simultaneously, and the equation (8) has a double root The equation n(x, y) = may therefore be said to rsptaaspl tbo locus of those points in the plane for which two of the eurrsf of the given family which pass through it have merged into a single one. The figures 37, a, and 37, A, show clearly the manner in wkieh two of the curves through a given point merge into a single one as that point approaches a point of the eunre if(«, ]f ) » 0| whotlior OS the true envelope or on a lot'us t»f double points. 430 PLANE CURVES [X,§203 Note. It often becomes necessary to find the envelope of a family of curves (9) F{x, y, a, ^) = whose equation involves two variable parameters a and 5, which themselves satisfy a relation of the form <^(a, b) = 0. This case does not differ essentially from the preceding general case, however, for 6 may be thought of as a function of a defined by the equation ^ = 0. By the rule obtained above, we should join with the given equation the equation obtained by equating to zero the derivative of its left-hand member with respect to a : da db da But from the relation <^(a, 6) = we have also d^ d<f> db _ da db da ' whence, eliminating db/da, we obtain the equation ^ ^ ^ ^^"^ da db db da " "' which, together with the equations F = and </> = 0, determine the required envelope. The parameters a and b may be eliminated between these three equations if desired. 203. Envelope of a straight line. As an example let us consider the equation of a straight line D in normal form (11) xcosa + ysinar -/(a) = 0, where the variable parameter is the angle a. Differentiating the left-hand side with respect to this parameter, we find as the second equation (12) — xsina + ycosa — /(a) =0. These two equations (11) and (12) determine the point of intersection of any one of the family (11) with the envelope E in the form . jgv ( X = f(a) cos a - f'{a) sin a , j y = f{a) sin a + f{a) cos a . It is easy to show that the tangent to the envelope E which is described by this point (z, y) is precisely the line D. For from the equations (13) we find /14V ( dx = - [/{a) ■\-ria)] sinorda, ^ ' ldy= [/(a) 4- /"(a)] cos a dor, whence dy/dx = — cot a , which is precisely the slope of the line D. X.$203] ENVKLOPE8 ttl Moreover, If g denote the length of the an of ih» MvvloDe horn «•• a—j point upon it, we hare, from (U), ^^ ^^ ^ "•• • = ± [//(a) da +/'(«) J. Hence the envelope will be a curve which to eadly racUflable If we for/(a) the derivative of a known function.* As an example let ua set /(a) = / sin a coe a. Tkklng ysOaods.OMA. cessively in the equation (11), we find (Fig. 88) O^ s /atoa OBmlcml respectively ; hence AB = I. The required ' curve is therefore the envelope of a atraight line of constant length /, whoee extremiUee always lie on the two axes. The formuUe (13) give in this case « = isin»a, y = /coe«a, and the equation of the envelope la (;)-(0^= which represents a hypocycloid with four cusps, of the form indicated in the ti/^ure. As a varies from to «'/2, the point of con- tact describes the arc DC. Hence the length of the arc, counts fmm D, to Flo. as =X'« alnaooaada 8/ — ( S Let / be the fourth vertex of the rectangle determined bj OA tad OA, tDd Jf the foot of the perpendicular let fall from / upon AB. Tliia, troa ll» tri- angles AMI and APM, we find, snoceMlvely, ^3f = ^/coea = <oo8*a, AP = AMtina es leo^atHma. Hence OP = OA - AP = t sin'a, and the point if to th« poiat ol the line AB with the envelope. Moreover hence the length of the are DM s ZBM/%. • Each of the quantities whieh ooeur In the fotvola lor «, $*f(a) 4 //to) 4a, has a geometrical meaning : a U the aagto ben reea the « axto aad the peipaiiSMlar OX let fall upon the variable line from the orifhi: /{a) to the ittotaare OS tmm lie origin to the variable line; and /'(<t) to, eJBBCpt for riga, the 4toiaawe JfJf itmm the point }f where the variable line tooebei Its envelope to the Inol Jfof the (liiMilar let fall upon the line from the origin. The feramhi lor • to effeei Legendrt's formula. 432 PLANE CURVES [X,§204 Fio. 39 804. Snrelope of a circle. Let us consider the family of circles (16) (x-a)a+(y-6)2-p^ = 0, where a, 6, and p are functions of a variable parameter t. The points where a circle of this family touches the envelope are the points of intersection of the circle and the straight line (16) (X - a) a' + (y - b) 6' + pp' = 0. This straight line is perpendicular to the Ungent MT to the curve C described by the center (a, 6) of the variable circle (16), and its distance from the center is p dp/ds, where s denotes the length of the arc of the curve C measured from some fixed point. Consequently, if the line (16) meets the circle in the two points N and N\ the chord NN' is 'c/ bisected by the tangent MT at right angles. It follows that the envelope y^ ^\ ^.^-'-yf^ I /' consists of two parts, which are, in ^"^ ^ ' general, branches of the same analytic curve. Let us now consider several special cases. 1) If /o is constant, the chord of con- tact JVjV' reduces to the normal PP^ to the curve C, and the envelope is com- posed of the two parallel curves Ci and a which are obtained by laying off the constant distance p along the normal, on either side of the curve C. 2) If p = a + iT, we have p dp/ds — p, and the chord l^W reduces to the tan- gent to the circle at the point Q. The two portions of the envelope are merged into a single curve r, whose normals are tangents to the curve C. The curve C is called the enolnte, of r, and, conversely, r is called an involute of C (see § 206). If dp>d8j the straight line (16) no longer cuts the circle, and the envelope is imaginary. 8ec(mdary caustics. Let us suppose that the radius of the variable circle is propor- tional to the distance from the center to a fixed point O. Taking the fixed point as the origin of coordinates, the equation of the circle becomes (« - a)* + {y- 6)« = lfl{a* + fta), where I: is a constant factor, and the equation of the chord of contact is (« - a)a' + (y - b)l/ + k*{aa' + WO = 0. If 8 and S' denote the distances from the center of tlie circle to the chord of contact and to the parallel to it through the origin, rcipectlvely, the preceding equation shows that d = k^S'. Let P be a point on the radius MO (Fig. 40), such that MP = k^MO, and let C be the Fig. 40 X»53W] CURVATURE U$ locus of the center. Then the eqiuukm Joit f*« iiH ^mwi thtt IW flteid el qm tact U the perpendionUr let lall from P upon the tenieiii to C ftlihe mmim M Let ua suppoee tb*t ic in le« than unity, &nd let K denote that bcBneh of the envelope which Hee on the lame tide of the t«nfent MT ee doee Um potet a Let i and r, respectively, denote the two angles which the two Ums MO Md liN make with the normal JT/ to the corre C. Then w «if| | ^f^ 8ini = ^. rinr = i^. !*?^' = :?!>« IS „ » Now let U8 imagine that the point O Ii a eooroe of light, and that the ■eparates a certain homogeneoue medium in which O lies from ai _ whose index of refraction with respect to the first is \/k. At\mt r«frac«kMi the incident ray OM will be turned Into a refracted ray MR^ which, hy the law of refraction, is the extension of the line NM. Hence all the ivfracted rays Jlffi are normal to the envelope, which is called the mcondary eamMte ai rafraecloa. The true caustic, that is, the envelope of the refracted raja, te the avotale of Ika secondary caustic. The second branch E' of the envelope evidently has no pbyileal it would correspond to a negative index of refraction. If we set ft « 1, envelope E reduces to the single point O, while tlie portion JT hofimw the loew of the points situated symmetrically with with respect to the taofniu to C. This portion of the envelope is also the secondary caustic qf r^ftedfam for Ind- dent rays reflected from C which issue from the fixed pobit O. It may he shows in a manner similar to the above that if a circle he described aboat each poiatsi C with a radius proportional to the distance from its center to a isid slcri^tt line, the envelope of the family will be a seoondary canstie with reipsel to a system of parallel rays. n. CURVATURE 205. Radius of curvature. The first idea of oairatare it that Um curvature of one curve is greater than that of another if it more rapidly from its tangent In order to render thia vague idea precise, let us first consider the caae of a eirele. Its curvature increases as its radius diminishes; ii is therefore quilt natural to select as the measure of itt curratore the timplett fvae- tion of the radius which increatet at the radint diminiahet, that is, the reciprocal 1/R of the radiut. Let i4i} be an are of a eirek of radius R which subtends an angle m at the oentar. The aafla between the tangents at the extremities of the are ^B is alto«, aod the length of the arc is « = Ru. Hence the measure of the earra- ture of the circle is w/a. This last definition maj be esteadsd lo an arc of any curve. Let .4 B be an aro of a plane eurre without a point of inflection, and •* the angle between the tangent* at tha extremities of the arc, the directions of the tansenta baiiif talWi in the same sense according to some rule, — tha diftelioB froHi A y r Bf ^ /j. -^J^ ^ — :?^ X 434 PLANE CURVES [X,§206 toward J5, for instance. Then the quotient <u/arc AB\^ called the average Curvature of the arc AB. As the point B approaches the point A this quotient in general approaches a limit, which is called the curvature at the point A. The radius of curvature at the point ^ is defined to be the radius of the circle which would have the same curvature which the given curve has at the point A\ it is therefore equal to the recipro- cal of the curvature. Let s be the length of the arc of the given curve measured from some fixed point, and ^°' a the angle between the tangent and some fixed direction, — the a? axis, for example. Then it is clear that the average curvature of the arc ^i5 is equal to the absolute value of the quotient Aa/As ; hence the radius of curvatui'e is given by the formula R = ± lim — = ± -r-- Acr da Let us suppose the equation of the given curve to be solved for y in the form y =f(x). Then we shall have a = arc tan y\ da = -r^ -i ds = Vl -\- y'^dx, and hence (17) ie=±ii±p!. Since the radius of curvature is essentially positive, the sign ± indicates that we are to take the absolute value of the expression on the right. If a length equal to the radius of curvature be laid off from A upon the normal to the given curve on the side toward which the curve is concave, the extremity / is called the center of curvature. The circle described about / as center with R as radius is called the circle of curvature. The coordinates (xq,. y^ of the center of curvature satisfy the two equations (1 + yT which express the fact that the point lies on the normal at a dis- taDce R from A. From these equations we find, on eliminating x^ (Xj - x) + {y, - y)y' = 0, (x, - xy + {y, - yY X,5306] CURVATURE 435 In order to tell which sign should be taken, lei 111 note tluit if /* is positive, as in Fig. 41, y^-y must be positive; henee the potttiTe sign should be taken in this case. If y" is negatiTe, yi — y is neO' tive, and the positive sign should be taken in this oaae ^h wi TIm coordinates of the (>eni(>r of curvatare are therefore given bv tbi formulae (18) y.-. = ^'. x.-.-yi±J!lV When the co5rdinate8 of a point (x, y) of the variable curve ate given as functions of a variable parameter t^ we have, hy f 3S, '' dx ^ da* and the formulae (17) and (18) become (19) dx d*y — dy rf*x dxd'y-dyd^x ^* ^ dxd^y-dyd^x At a point of inflection y" = 0, and the radius of eorvatars b infinite. At a cusp of the flrst kind y can be developed aeooiding to powers of x^'^ in a series which begins with a term in x ; kaoea y' has a finite value, but y" is infinite, and therefore the radios of curvature is zero. Noit. When the coordinates are expreved m foaetkas of Um are t of iIm curve, x = ^(»). If = f(»). the functions and f satisfy the relation 0'«(s) + ^«(«) = l. since dx^ \ d\f^ - dt^, and hence they also satisfy the rtfaUloo ^V + rf = 0. Solving these equations for ^* and f , wt find where « = ± 1, and the formula for the radius of owtataft tak« 00 U»» cially elegant form (20) ^= [«•)!• +[r(«)r. 436 PLANE CURVES [X,§206 206. Evolutes. The center of curvature at any point is the limit- ing position of the point of intersection of the normal at that point with a second normal which approaches the first one as its limiting position. For the equation of the normal is where X and Y are the running coordinates. In order to find the limiting position of the point of intersection of this normal with another which approaches it, we must solve this equation simulta- neously with the equation obtained by equating the derivative of the left-hand side with respect to the variable parameter x, i.e. The value of F found from this equation is precisely the ordinate of the center of curvature, which proves the proposition. It follows that the locus of the center of curvature is the envelope of the normals of the given curve, i.e. its evolute. Before entering upon a more precise discussion of the relations between a given curve and its evolute, we shall explain certain con- ventions. Counting the length of the arc of the given curve in a definite sense from a fixed point as origin, and denoting by a the angle between the positive direction of the x axis and the direction of the tangent which corresponds to increasing values of the arc, we shall have tan a =±y', and therefore ,1 .dx cos a = ± , — = ± -!-• Vl -f y'^ ds On the right the sign -f- should be taken, for if x and s increase simultaneously, the angle a is acute, whereas if one of the varia- bles X and 8 increases as the other decreases, the angle a is obtuse ($ 81). Likewise, if p denote the angle between the y axis and the tangent, cos p = dy/ds. The two formulae may then be written dx . dy C08a = -;-> sina = -p-> ds ds where the angle a is counted as in Trigonometry. On the other hand, if there be no point of inflection upon the given arc, the positive sense on the curve may be chosen in such a way that 8 and a increase simultaneously, in which case R = ds/da all along the arc. Then it is easily seen by examining the two possible cases in an atrtual figure that the direction of the segment X,fa08] CURV ATI-RE 4S7 starting at the point of the curve and going to the ocnter of etam. ture makes an angle ai = a-^ ir/2 with the x axis. The ^>i.i> H f|n|„ (^i, yi) of the center of curvature are therefore given by th» ibnmOs x, = x-h R coe(a' -f ^j = X - /? ain a, yi = y-H/« sin ^a-f I) «y + iJooea, whence we find dxi = coaads — Rco8 a da— Bin a dR^-~ gin a dJt, di/i=s Bin ads — R sin a da -^ COB a dH» oOBadR, In the first place, these formulae show that dy^/dx^ ^ — efAa, which proves that the tangent to the evolute is the normal to the giveo curve, as we have already seen. Moreover or rf«i = ± dR. Let us suppose for definiteness that the radius of curvature constantly increases as we proceed along the cnrre C (Fig. 42) from 3/^ to 3/^, and let us choose the poaitiTii mmm of motion upon the evolute (/)) in such a way that the arc ^i of (Z)) increases simultane- ously with R. Then the preceding formula becomes ds^ = dRy whence *i = i2 + C. It follows that the arc /i l^ — R^ — /?i, and we see that the length of any are of the evoluts is equal' to the difference between the two radii of curvature of the curve C whiek eoT' respond to the extremities of that arc. This property enables us to construct the involute C mechanically if the evolute (D) be given. If a string be attached to (D) at an arbitrary point A and rolled around (£>) to /«, thence following the tangent to A/,, the point ^ft will deacribe the inTolnto C at tiie string, now held taut, is wound further on roond (/>). This ooo* struction may be stated as follows : On each of the tangents iU of the evolute lay off a distance IM = I, where / + « » oontt, « being the length of the arc /< / of the evolute. Avigning Tarioot valnee to the constant in question, an infinite number of involutes may be drawn, all of which are obtainable from any one of Umb by lajing off constant lengths along the normals. 438 PLANE CURVES [X, §207 All of these properties may be deduced from the general formula for the differential of the length of a straight line segment (§ 82) dl = — da-i cos (Oj — dcr^ COS <i>2 . If the segment is tangent to the curve described by one of its extremities and normal to that described by the other, we may set ,oi = TT, ci>2 = 7r/2, and the formula becomes dl — da-i = 0. If the straight line is normal to one of the two curves and I is constant, dl = 0, cos Oil = Oj and therefore cos wj = 0. The theorem stated above regarding the arc of the evolute depends essentially upon the assumption that the radius of curvature con- stantly increases (or decreases) along the whole arc considered. If this condition is not satisfied, the statement of the theorem must be altered. In the first place, if the radius of curvature is a maxi- mum or a minimum at any point, dR = at that point, and hence dxi = dyi = 0. Such a point is a cusp on the evolute. If, for example, the radius of curvature is a maximum at the point M (Fig. 43), we shall have arc//i = /JW-/iMi, arc 7/2 = IM — I2M2J whence arc /1//2 = 2IM — I^M^ — I^M^. Hence the difference I^M^ — I^M^ is equal to the difference between the two arcs II^ and II^ and not their sum. 207. Cycloid. The cycloid is the path of a point upon the circumference of a circle which rolls without slipping on a fixed straight line. Let us take the fixed line m the x axis and locate the origin at a point where the point chosen on the circle lies in the x axis. When the circle has rolled to the point / (Fig. 44) the point on the circumference which was at O has come into the position Af, X. $ 207] CURVATURE 4i9 where the circular arc /If i« equal to the n^imiiu OJ, Lai i between the raUii CM aud CI an the rariable parameier. Tbao Iba of the poiut if are x= 0/-/P = H^-Baln^, v = J>1^ ^ IC -^ CQ^ R - Hvm^, where P and Q are the proJeoUona of if on the two Uaca 01 and /T, tively. It is easy to show that these fonnulai hold for any ralaa of tiM aofk #w In one complete revolation tlie point whose path Is soocbt OBOi . If the motion be continued indeflniiely, we obtain an of arcs congruent to this one. From the praoedlng formuhe we find X = ii(0 - sin 0) , dx = R{\-cM^)d4, iPzs R^n^d4^, y = £(l-cos^), dy = Ruin^d^, #lf s Aoos^df*, and the slope of the tangent is seen to be dy tin^ ,» — - = — = cot-» dz l--cos^ a which shows that the tangent at if la the ttntlght line if T, since tbe angk MTC = <p/2, the triangle if rc being UMisceles. Hence the normal at if is tW straight line MI through the point of tangency / of the fixed straight line with the moving circle. For the length of the arc of the eyclold we find ds'i = Ii^d<p^[s\n^<p-i-{\-coB^)*]=iR*ain*^d4^ or dssil{aia>4#. if the arc be counted in the sense in which It increaaea with ^ the arc from the point as origin, we shall have =4ii(i-co.|y Setting = 2sr, we find that the length of one whole aeetton OBOi la tR. Tha length of the arc 0MB from the origin to the m a ximnm B la tharafore 4R, aad the length of the arc BM (Fig. 44) is 4R coa#/2. From the triaagia if TX7 Iha length of the segment if T is 2R coe^/2 ; hence arc Bif = 2if T. Again, the area up to the ordinate throng if Is A =f*ydx =/*R»0 - «ooa^ + oot^f)d^ 2coa^ + /3 « t . ain2#\ = (-0-2sin^ + -^) Hence the area bounded by the whole arc OBQx sii'i the base OOi » a««% ins4 is, three times the area qf the genaraUng eircU, (GAUtao.) ^^^ The formula for the radius of cunrature of a plana tnrm glvw lor the cycMd 8/J«sin«^d#« o= ^ — = 4Raln?. aJPsln«|d^ 440 PLANE CURVES [X, § 208 On the other hand, from the triangle MCI, MI = 2R sin 0/2. Hence p = 2MI, and the center of curvature may be found by extending the straight line Ml past / by its own length. This fact enables us to determine the evolute easily. For, consider the circle which is symmetrical to the generating circle with respect to the point I. Then the point M' where the line MI cuts this second circle is evidently the center of curvature, since M'l = MI. But we have arc T'M' = tcR - arc IM' = TtR - arc IM = xR - 01, or arc T'Jf' =OH-OI = IH= T'R. Hence the point M' describes a cycloid which is congruent to the first one, the cusp being at B' and the maximum at 0. As the point M describes the arc BOi , the point 3f ' describes a second arc B'Oi which is symmetrical to the arc OB" already described, with respect to BB". 208. Catenary. The catenary is the plane curve whose equation with respect to a suitably chosen set of rectangular axes is (21) y = |^e5 + e-5V Its appearance is indicated by the arc MAM' in the figure (Fig. 45). y From (21) we find If denote the angle which the tangent TM makes with the x axis, the formula for / gives Bin<f> e" — e <* 1 Z' COS0 e« -f- e <* The ndios of curvature is given by the formula r a' But, Id the triangle MPN, MP = JlfJVcos ; hence COB0 (t X. $ 209] CURV ATURB 44 1 It followH that the radim of eiinratar« of the eummry !■ eqaal to iIm Imgtk •# the normal MN. The etolnte may be found witboot dUBcolty fraa Uito *tf* The length of the arc AM of the catenary la giran by tbo ■r=¥-''-i(-'--') or a = V sin 0. If a perpendicular Fm ba dropped froBi i' ^i-ig. 4oj upoa i|m tangent if 7, we tind, from the triangle Pifm, jrm = Jn>aln^r=a. Hence the arc ^Jf is equal to the dJitanco Mm. 209. Tractrix. The curve described by the point m (FIf. 4^ Is oUlsd lbs tractrix. It is an involute of the catenary and has a eosp at lbs point A, Tbs length of the tangent to the tractrix is the distance mP. But, in the iftaafis MPm, mP = y cos = a ; hence the length mP measorsd akwg tbs lasfMl to the tractrix from the point of tangency to the x axJs is ronitint and sqaal to «. The tractrix is the only curve which hss this property. Moreover, in the triangle MTP, Mm x mT = d*. Hence tbs prodoet of tbs radius of curvature and the normal is a constant for tbs tractiiz. Tbiipioponj is shared, however, by an infinite number of other pUns cartes. The coordinates (xi, yi) of the point m are given by tbs e« - « • Xi = x — soos^ = g — a , S* + f"« So yi = y-«8in^ = -^^ -, ei + «"« or, setting e'/o = tan 9/2, the equations of the tractrix ars log(tan?), (22) xi = ocoe^ + alogitan- 1, v. -aninJ. As the parameter varies from ir/2 to ir, the point lyj . yii aracnoMi tarn mrv Avin, approaching the x axis as asymptote. As # varies (roa «/S to A, Ibo point (xi , vi) describes the arc Am'n\ qrmmstrloal to tbs flat witb rsspsei to the y axis. The arcs Amn and Am'n' oorrespoiid, wqwcllfsty, to tbs arcs AM and AM' of the catenary. 210. Intrinsic equation. Let us try to curve when the radius of curvature R Is U = 0(s). Let or be the angle betwoan Rsz± ds/da, and therefore A first integration givet X*i^ a = a,± . ^ 442 PLANE CURVES [X,§210 and two further integrations give x and y in the form x = Xo+ / C08ad«, y = yo + r sinada. The curves defined by these equations depend upon the three arbitrary con- stants Xo, Voi and oq. But if we disregard the position of the curve and think only of its form, we have in reality merely a single curve. For, if we first con- sider the curve C defined by the equations the general formulae may be written in the form X = Xo + -^cos ao — FsinoTo, y = yo + -2" sin aro + Fcosoro, if the positive sign be taken. These last formulae define simply a transforma- tion to a new set of axes. If the negative sign be selected, the curve obtained is symmetrical to the curve C with respect to the X axis. A plane curve is therefore completely determined, in so far as its form is concerned, if its radius of curvature be known as a function of the arc. The equation R = <p{s) is called the inirinsic equation of the curve. More generally, if a relation between any two of the quantities JS, s, and a be given, the curve is completely deter- mined in form, and the expressions for the coordinates of any point upon it may be obtained by simple quadratures. For example, if R be known as a function of a, R=f{a)^ we first find d» = f{a) da, and then dx = cosa/(a:)da, dy= sin. a f (a) da f whence x and y may be found by quadratures. If i^ is a constant, for instance, these formulas give X = Xo + iJ sin a , y = yo — R cos a , and the required curve is a circle of radius R. This result is otherwise evident from the consideration of the evolute of the required curve, which must reduce to a single point, since the length of its arc is identically zero. As another example let us try to find a plane curve whose radius of curva- ture is proportional to the reciprocal of the arc, R = ays. The formulae give and then JC'ads «« ' — = — » Although these integrals cannot be evaluated in explicit form, it ia easy to gain an idea of the appearance of the curve. As s increases from to + oo, x and y each pass through an infinite number of maxima and minima, and they approach the same finite limit. Hence the curve has a spiral form and approaches asymptotically a certain point on the line y = x. X,52n] CONTACT OSCULATION m. CONTACT OF PLANE CURVES 211. Order of conuct. Let C and C be two plana eurrct whkb are tangent at some point A. To every point m on C lei ua uptifn, according to any arbitrary law whatever, a point m' on C, the only requirement being that the point m' should approach A with m. Taking the arc A in — or, what amounts to the same thing, the chord Am — as the principal infinitesimal, let us first investigate what law of correspond- ence will make the order of tlie infin- itesimal 7n7n' with respect to i4m as large as possible. Let the two curves be referred to a system of rectangular or oblique cartesian coordinates, the axis o/jfnai being jMralUi to tks common tangent A T. Let (C) >' = n*) be the equations of the two curves, respectively, and let (j^, m) be the coordinates of the point A, Then the coordinates of m will be [a^o + K /(^o + A)], and those of m' will be [ac^ -h A, F{x^ -»- Ir)], where k is a function of A which defines the ooirespondence be i n-eeo the two curves and which approaches zero with h. The principal infinitesimal Am may be replaced by A S3 ap^ ioit the ratio 'a/>/.l7/i approaches a finite limit different from sero as the point m approaches the point A, Let us now suppose that mm' is an infinitesimal of order r -f 1 w ith r espect to A, for a certain method of correspondence. Then mw? is of 0fd«r 2r 4- 2. If # denote the angle between the axes, we shall have ^;^'' = [F(2-„ + A:) -fix. + A) + (ife - A) coarp + (* - A)«sin«#; hence each of the differences k — k and F{x^ + k) —/{x^ + A) mnst be an infinitesimal of order not less than r + 1, that ia, ;fc = A H- aA"-**, F(x. + *) -/t't + *) - fi^^^'f where a and p are functions of A which approach finite limits as A approaches zero. The second of these formolm may be written in the form F(xo + A -f aA"^») -yt'. + *) « ^A'**. 444 PLANE CURVES [X,§2ii If the expression F(xq-{- h + ah"-*-^) be developed in powers of a, the terms which contain a form an infinitesimal of order not less than r -h 1. Hence the difference A = F(x, + A) -f(x, 4- h) is an infinitesimal whose order is not less than r-\-l and may exceed r -f 1. But this difference A is equal to the distance mn between the two points in which the curves C and C are cut by a parallel to the 1/ axis through m. Since the order of the infinitesimal mm' is increased or else unaltered by replacing m' by n, it follows that the distance between two corresponding points on the two curves is an infinitesimal of the greatest possible order if the two corresponding points always lie on a parallel to the y axis. If this greatest possi- ble order is r -f- 1, the two curves are said to have contact of order r at the point A. Notes. This definition gives rise to several remarks. The y axis was any line whatever not parallel to the tangent A T. Hence, in oi-der to find the order of contact, corresponding points on the two curves may be defined to be those in which the curves are cut by lines parallel to any fixed line D which is not parallel to the tan- gent at their common point. The preceding argument shows that the order of the infinitesimal obtained is independent of the direc- tion of /), — a conclusion which is easily verified. Let mn and mm^ be any two lines through a point m of the curve C which are not parallel to the common tangent (Fig. 46). Then, from the triangle fH7n, fif mm mn sin mm' n As the point m approaches the point A, the angles mnm' and mm'n approach limits neither of which is zero or tt, since the chord m'n approaches the tangent AT. Hence mm'/mn approaches a finite limit different from zero, and mm' is an infinitesimal of the same order as mn. The same reasoning shows that mm' cannot be of higher order than mn, no matter what construction of this kind is used to determine m' from 7n, for the numerator sin m7im' always approaches a finite limit different from zero. The principal infinitesimal used above was the arc ^m or the chord Am. We should obtain the same results by taking the arc An of the curve C for the principal infinitesimal, since Ani and An are infinitesimals of the same order. X,§212] CONTACT OSCULATION 446 If two curves C and C have a contaet of order r, the poioU m* on C may be assigned to the pointe m on C in an infinite ntinikMr of ways which will make mm' an infiniteaimai of order r 4- 1, — fur that purpose it is sufficient to tet k ^ h-^ ah'**, where #^ r nod where a is a function of A which remains finite for A b 0. On the other hand, if « < r, the order of mm' cannot exceed « 4- 1. 212. Analytic method. It follows from the preceding seeiioo thai the order of contact of two curves C and C* is given by evaluating the order of the infinitesimal y-y = F(x, + A)--y(T, + A) with respect to h. Since the two curves are tangent at A, /\xo) =f(xo) and F'(xo) =/'(3?o)- It may happen that others of the derivatives are equal at the same point, and we shall suppose for the sake of generality that this is true of the first n derivatives : r>3^ \ F(x„)=f{x,), FXx,)^r(',), but that the next derivatives f '•♦"(!,) and /••♦ "(i,) aw nnaqaaL Applying Taylor's series to each of the functions F(x) aad/{xf, w« find or, subtracting, (24) >^-y = i.2.^-(l-n) ^^'*"^'^^--^'"''^'^^'"*^' where c and e' are infinitesimals. H/oUowm tAmi tMe^fdermfi of two ctirves is equal to the orrfw n 0/ ths higheti derivaii9t» ^f Fi^z) and fix) which are equal for x = x,. The conditions (23), which are due to Lagrange, are the neeeesaiy and sufficient conditions that ac « x^ should be a multiple root of order n -f 1 of the equation F(x) =/(x). But the rooto of thit equation are the abecissn of tho iminu of intersection of the two 446 PLANE CURVES [X, §212 curves C and C ; hence it may be said that two curves which have contact of order n have n -{• 1 coincident points of intersection. The equation (24) shows that Y — y changes sign with A if ti is even, and that it does not if n is odd. Hence curves which have contact of odd order do not cross, but curves which have contact of even order do cross at their point of tangency. It is easy to see why this should be true. Let us consider for definiteness a curve C' which cuts another curve C in three points near the point A. If the curve C" be deformed continuously in such a way that each of the three points of intersection approaches A, the limiting position of C has contact of the second order with C, and a figure shows that the two curves cross at the point A. This argument is evidently generaL If the equations of the two curves are not solved with respect to Y and y, which is the case in general, the ordinary rules for the calculation of the derivatives in question enable us to write down the necessary conditions that the curves should have contact of order n. The problem is therefore free from any particular diffi- culties. We shall examine only a few special cases which arise frequently. First let us suppose that the equations of each of the curves are given in terms of an auxiliary variable ( x=f(t), iX=^f(u), (C) \ -^^^^ (C) \ -^^ ^' and that \f/(to) = </>(^o) ^^^ ^'(^o) — <^'(^o)j i-^- t^3,t the curves are tan- gent at a point A whose coordinates are f(t(i), <t>(to). Iff'{t^ is not zero, as we shall suppose, the common tangent is not parallel to the y axis, and we may obtain the points of the two curves which have the same abscissae by setting u = t. On the other hand, x — Xq\q of the first order with respect to ^ — ^o> and we are led to evaluate the order of i/^(^) — <^(^) with respect tot — tQ, In order that the two curves have at least contact of order n, it is necessary and sufficient that we should have (26) ^^,{t,) = <A(^o) , •A'(^o) = <i>\to), • • • , lA^"^ (« = <^^"^ (^o) , and the order of contact will not exceed n if the next derivatives ^•■^'>(^) and <^<'' + '>(^o) are unequal. Again, consider the case where the curve C is represented by the two equations (26) a^'/lO. y = *(0. X, i 212] CONTACT OSCULATION 447 and the curve C by the single equatioD F(x, y) ■■ 0. This cam nuiy be reduced to the preceding by replacing » in F(m^ y) by /(I) and considering the implicit function y a ^(t) defined by the equaltoo (27) F[/(<),f(0]-0. Then the curve C is also represented by two eqoaliona of Um (28)* *=/(0, y = ^'V In order that the curves C and C should have contact ol Older m at a point A which corresponds to a value t^ of the parameter, it is necessary that the conditions (25) should be satisfied. But the successive derivatives of the implicit function ^t) are giTeo by the equations (29) d"F ^[A0/+ +X7^-'W Hence necessary conditions for contact of order n will be by inserting in these equations the relations f = f„ X =/(fo), ^*o) = ^(^o), f (« = ♦'(<•)» • • •» r'\t.) = ♦-'(!•). The resulting conditions may be expressed as foUowa : Let F(o=nAO» ♦{')]; then the two given curves teill have <U Utui eamiati ^ 0rdw if mmd only if (30) F(fo) = 0, F'(<.) = 0, p*»{g-o. The roots of the equation f{t) » are the valnee of I which cot^ respond to points of intersection of the two giTcn oonrea. Heaee the preceding conditions amount to saying thai « i- ^ ia a mvltiple root of order n, i.e. that the two ourree have a + 1 of intersection. 448 PLANE CURVES [X,§213 213. Osculating curves. Given a fixed curve C and another cm-ve C which depends upon n + 1 parameters ay bf Cj •• -, l, (31) F(Xyy, a, b,cy-,r) = 0, it is possible in general to choose these n -{- 1 parameters in such a way that C and C shall have contact of order n at any preassigned point of C. For, let C be given by the equations x =f(t), y = <f>(t). Then the conditions that the curves C and C should have contact of order n at the point where t = tQ are given by the equations (30), where ^(t) = F[f(t),<f>(t),a,b,c,--',l^. If Iq be given, these n -fl equations determine in general the n+l parameters a, b, c, •••, /. The curve C obtained in this way is called an osculating curve to the curve C. Let us apply this theory to the simpler classes of curves. The equation of a straight line y = ax -{- b depends upon the two param- eters a and b ; the corresponding osculating straight lines will have contact of the first order. If y =f{x) is the equation of the curve C, the parameters a and b must satisfy the two equations f(xo) = aXo-hb, f(x,) = a', hence the osculating line is the ordinary tangent, as we should expect. The equation of a circle (32) {x - ay +(y-by-R^ = depends upon the three parameters a, b, and R ; hence the corre- sponding osculating circles will have contact of the second order. Let y =f(x) be the equation of the given curve C ; we shall obtain the correct values of a, b, and R by requiring that the circle should meet this curve in three coincident points. This gives, besides the equation (32), the two equations (33) x-a + (y- b)y' = 0, 1 + y^ + (y _ b)y" = 0. The values of a and b found from the equations (33) are precisely the coordinates of the center of curvature (§ 206) ; hence the oscu- lating circle coincides with the circle of curvature. Since the con- tact is in general of order two, we may conclude that in general the circle of curvature of a plane curve crosses the curve at their point of tangency. X.f213] CONTACT OSCULATION 449 All the above results might hare been foraum • fH&ri, For, since the coordinates of the center of cumlUM dnMad miIt ^ X, y, y'y and y", any two curves which have oooteet of IIm f^n td order have the same center of curvature. But the eeoter of tur^ ture of the osculating circle is evidently the center of thai eirato itself; hence the circle of curvature most eoinotde witk tlie oms* lating circle. On the other hand, let us consider two circl« of curvature near each other. The difference between their radii, which is equal to the arc of the evolute between the two <*^tttiftw, is greater than tlie distance between the centers; hence one of the two circles must lie wholly inside the other, which eoold not happen if both of them lay wholly on one side of the eurra C in the neighborhood of the point of contact. It follows that tbay cross the curve C There are, however, on any plane curve, in general, certain pointa at which the osculating circle does not cross the curve; this ezcep* tion to the rule is, in fact, typical. Given a curve C which depoada upon n + 1 parameters, we may add to the n + 1 equations (SO) tba new equation provided that we regard t^ as one of the unknown qnantitiea determine it at the same time that we determine the ay b, Cj 'y I. It follows that there are, in general, on any plana curve C, a certain number of points at which the order of oon- tact with the osculating curve C is it + 1. For example, there ara usually points at which the tangent has contact of the aeoood order; these are the points of inflection, for which y" « 0. In ordar to find the points at which the osculating circle haa oontaot of the tliird order, the last of equations (33) must be differentiated again, whieli gives 3y'y" + (y-*)y"' = 0, or finally, eliminating y — by (34) (l + y'«)y"'-3yy = 0. The points which satisfy this last condition are thoaa for which dR/dx = 0, i.e. those at which the radius of eorvatora ia a maxi- mum or a minimum. On the ellipse, for example, tbeae poinU are the vertices ; on the cycloid they are the poinU at which the tan- gent is parallel to the base. 450 PLANE CURVES [X,§214 214. Osculating curves as limiting: curves. It is evident that an osculating curve may be thought of as the limiting position of a curve C which meets the fixed curve C in n-^1 points near a fixed point A of C, which is the limiting position of each of the points of intersection. Let us consider for definiteness a family of curves which depends upon three parameters a, b, and c, and let ^0 + ^1 » ^0 + ^a> a.nd to + As be three values of t near t^. The curve C which meets the curve C in the three corresponding points is given by the three equations (35) F(^o + Ai) = , F(^o 4- ^2) = , F(^o + ^3) = . Subtracting the first of these equations from each of the- others and applying the law of the mean to each of the differences obtained, we find the equivalent system (36) F(fo + ^) = , F'(^o -\-h) = 0, F(t, + A:^) = , where ki lies between hi and Agj and kz between hi and h^. Again, subtracting the second of these equations from the third and apply- ing the law of the mean, we find a third system equivalent to either of the preceding, (37) F(to-hhi) = 0, F'(^o + A;i) = 0, F"(^o + ^i)=0, where Zi lies between ki and kz- As hi, h^, and h^ all approach zero, ki, k^j and li also all approach zero, and the preceding equa- tions become, in the limit, F(^o) = 0, F'(^o) = 0, F"(^o) = 0, which are the very equations which determine the osculating curve. The same argument applies for any number of parameters whatever. Indeed, we might define the osculating curve to be the limiting position of a curve C which is tangent to C at jo points and cuts C at q other points, where 2p + g' = n + 1, as all these p -\- q points approach coincidence. For instance, the osculating circle is the limiting position of a circle which cuts the given curve C in three neighboring points. It is also the limiting position of a circle which is tangent to C and which cuts C at another point whose distance from the point of tangency is infinitesimal. Let us consider for a moment the latter property, which is easily verified. Let us take the given point on C as the origin, the tangent at that point as the x axis, and the direction of the normal toward the X,Ex..] EXERCI8E8 461 center of curvature as the positiTe direetion of tli« y axi«. At tiM origin, y' = 0. lience R = 1/y", and theraforo, by Taylor't werim. y^^'fe-*-*)' where c approaches zero with x. It fol- lows that It is the limit of the expres- sion xV(2y) = 0P*/(2MP) at the point M approaches the origin. On the other hand, let Ki be the radius of the circle Ci which is tangent to the 2; axis at the origin and which passes through M, Then we shall have rio.«7 OP* = Mm^ = MP(2Rt - MP), or OP* 2AfP MP hence the limit of the radius 7?, is really equal tn Um radius of curvature R 1. Apply the general formul» to find the eTolale of sa elllpM : of sa hyper- bola ; of a parabola. 2. Show that the radius of cnrvsture of a conic Is pct^iorUoiuU lo lbs 4whs of the segment of the normal between its points of intsissdioii with tht ew<t and with an axis of symmetry. 3. Show that the radius of curvature of the psrshohi is eqoal to twiet ihe segment of the normal between the curve and the dlrselriz. 4. Let F and i?" be the foci of an eUipee, M s point oa tbo sUipss, MN Ihe normal at that point, and N the point of intersection of that aorauU sad the major axis of the ellipse. Erect s perpendicnlsr NK to MN st N, aiesfinf MF at K. At K erect a perpendicular KO to MF, meeting MN st O. O is the center of curvature of the ellipse at the point M, 6. For the extremities of the major axis the illusory. Let A OA' be the uMJor axis and BO^B the niaor axis of the On the segments OA and OB construct the rectaagle OA KB Frtm JT Ist tell a perpendicular on A B, meeting the major and minor axes st C snd iX raipeo* tively. Show that C snd D are the centers of curratare of the sUipas for llM pointj) A and B, respectively. 6. Show that the evolute of the spiral p s mr^ Is a given spiral. toihs 462 PLANE CURVES [X, Em. 7. The path of any point on the circumference of a circle which rolls with- out slipping along another (fixed) circle is called an epicycloid or an hypocycloid. Show that the e volute of any such curve is another curve of the same kind. 8. Let AB he an arc of a curve upon which there are no singular points and no points of inflection. At each point m of this arc lay off from the point m along the normal at m a given constant length I in each direction. Let mi and m« be the extremities of these segments. As the point m describes the arc AB, the points mi and ma will describe two corresponding arcs AiBi and ^2^2- Derive the formulas Si = S — ld, 82 = 8 + 16, where S, Si, and S2 are the lengths of the arcs AB, AiBi, and ^2 -^2 , respectively, and where 6 is the angle between the normals at the points A and B. It is supposed that the arc -^i^i lies on the same side of AB as the evolute, and that it does not meet the evolute. [Licence, Paris, July, 1879.] 9. Determine a curve such that the radius of curvatures p at any point M and the length of the arc s = AM measured from any fixed point A on the curve satisfy the equation ds = p^ ■{■ a^, where a is a given constant length. [Licence, Paris, July, 1883.] 10. Let C be a given curve of the third degree which has a double point at O. A right angle MON revolves about the point O, meeting the curve C in two variable points 3f and N. Determine the envelope of the straight line MN. In particular, solve the problem for each of the curves Xy2 = x^ and x^-\-y^ = itxy. [Licence, Bordeaux, July, 1885.] 11. Find the points at which the curve represented by the equations X = a (nw — sin ft>) , y — a{;n — cos w) has contact of higher order than the second with the osculating circle. [Licence, Grenoble, July, 1885.] 12. Let m, mi , and m2 be three neighboring points on a plane curve. Find the limit approached by the radius of the circle circumscribed about the triangle formed by the tangents at these three points as the points approach coincidence. 13. If the evolute of a plane curve without points of inflection is a closed curve, the total length of the evolute is equal to twice the difference between the sura of the maximum radii of curvature and the sum of the minimum radii of curvature of the given curve. 14. At each point of a curve lay off a constant segment at a constant angle with the normal. Show that the locus of the extremity of this segment is a curve whose normal passes through the center of curvature of the given curve. 16. Let r be the length of the radius vector from a fixed pole to any point of a plane curve, and p the perpendicular distance from the pole to the tangent. Derive the formula R = ± rdr/dp, where R is the radius of curvature. 16. Show that the locus of the foci of the parabolas which have contact of the second order with a given curve at a fixed point is a circle. 17. Find the locus of the centers of the ellipses whose axes have a fixed direc- tion, and which have contact of the second order at a fixed point with a given curve. CHAPTEK XI 8KSW CURVES I. OSCULATING PLANE 215. Definition and equation. Let AfThe the tangent at a point M of a given skew curve r. A plane through MT and a point i/' of r near M in general approaches a limiting position aa the point !#' approaches the point M. If it does, the limiting position of the plane is called the osculating plane to the curre F at the point i#. We shall proceed to find its equation. Let (1) ^=/(0. y = *(0» * = «0 be the equations of the curve T in terms of a parameter f, and let t and t -\- hhQ the values of t which correspond to the pointa J# and M\ respectively. Then the equation of the plane MTM* it ^(X - X) + B(r - y) + C(Z - «) = 0, where the coefficients Aj B, and C muat satiafy the two relatiooa (2) Af(t) + B4>\t)-\-Ci^(t)^0, (3) /l[/(?-hA)-y];0]+^W + A)-^(0]+C[f(r + A;-^tOj-0. Expanding f(t 4- h), ^{t + A) and ^(< -f A) by Tajlor'a aartea, tkm equation (3) becomes After multiplying by h, let us subtract from thia equation the eqaar tion (2), and then divide both sides of the reaulttng eqaatkm by A72. Doing 80, we find a system equivalent to (2) and (3): Aif"(t) + «o + B[r(t) + •.] + nrw + -] - 0, where c,, c„ and c, approach lero with A. In the limit aa k approaches zero the second of theae eqoatioiia (4) AfV) + ^♦"(0 + ^^W - ^ 4AS 464 SKEW CURVES [XI,§215 Hence the equation of the osculating plane is (6) A{X -x)-h B(Y-y) + C(Z-z) = 0, where A, B, and C satisfy the relations Adx -^Bdy -\- C dz =0, ^^^ \Ad^x + B d^y + Cd^z = 0. The coefficients A, B, and C may be eliminated from (5) and (6), and the equation of the osculating plane may be written in the form X-x Y-y Z-z dx dy dz d^x d^y d^z = 0. Among the planes which pass through the tangent, the osculating plane is the one which the curve lies nearest near the point of tan- gency. To show this, let us consider any other plane through the tangent, and let F{t) be the function obtained by substituting f{t -f h), <f>{t + h), \J/(t + h) for X, Y, Z, respectively, in the left-hand side of the equation (5), which we shall now assume to be the equa- tion of the new tangent plane. Then we shall have F(.t) = O f^-^"(*> + ^*"<^*) + ^"^"(') + ''^' where rj approaches zero with h. The distance from any second point M' of r near M to this plane is therefore an infinitesimal of the second order; and, since F(t) has the same sign for all sufficiently small values of h, it is clear that the given curve lies wholly on one side of the tangent plane considered, near the point of tangency. These results do not hold for the osculating plane, however. For that plane, Af" + B<l>" + C\f/" = ; hence the expansions for the coiirdinates of a point of T must be carried to terms of the third order. Doing so, we find ^W- 17273 V dg^ '^V' It follows that the distance from a point of V to the^ osculating plane is an infinitesimal of the third order ; and, since F({) changes sign with A, it is clear that a skew curve crosses its osculating plane at their common point. These characteristics distinguish the oscu- lating plane sharply from the other tangent planes. XI, §216] 08CULATlN(i I'LANK 455 216. Stationary otcalfttiiig pkne. The r«ralU Jntt obtaiatd an Boi valid if the coetiicieiits A^ Bf C of the oeculating pUoe mMkdw the relation (7) i4d««4-Brf«y + Crf««B0. If this relation is satisfied, the expansions for the oo6rdinatee muii be carried to terms of the fourth order, and we should obtain a relation of the form h< / Ad*x-^Bd*y^Cd*z \ ^^^^ 1.2.3.4V dt' ^V' The osculating plane is said to be stationary at any point of r for which (7) is satisfied ; if A d^x + Bd^y 4- C d^m does sot vaiiiah also, — and it does not in general, — F{t) changes sign with k and the curve does not cross its osculating plane. Moreover the distanee from a point on the curve to the osculating phuie at fuch a point ta an infinitesimal of the fourth order. On the other hand, if the relation A d*x -^ Bd*i/ -^ C d*x ^ is satisfied at the same point, the expansions would have to be carried to terms of the fifth order ; and so on. (7). we = 0, whose roots are the values of t which oorrespond to the points of f where the osculating plane is stationary. There are then, usually, on any skew curve, points of this kind. This leads us to inquire whether there are curves all of whose osculating planes are stationary. To be precise, let us try to find all the possible sets of three functions x, y, s of a single vaHahle #, which, together with all their derivatives up to and including those of the third order, are continuous, and which satisfy the equatioo (8) for all values of t between two limits atmdk(a<b). Let us suppose first that at least ooe of the minors of A which correspond to the elements of the third row, wkydxd^y-d^ ^ jr, does not vanish in the interval (a, h). The two equations W ( ,1. = Ctd'x + C,J'g, Eliminating A, By and C between the obtain the equation dx dy dx (8) A = d}x d^y d^x d^x d^y </•« 456 SKEW CURVES [XI, §216 which are equivalent to (6), determine Ci and Cg as continuous functions of t in the interval (a, b). Since A = 0, these functions also satisfy the relation (10) d^z = Cid^x + Cg d'^y. Differentiating each of the equations (9) and making use of (10), we find dC.dx + dC^dy = 0, dC^d^x + dC^d^y = 0, whence dCi = dC^ = 0. It follows that each of the coefficients Ci and Ci is a constant; hence a single integration of the first of equations (9) gives z = CiX + C^y-\'C^y where C, is another constant. This shows that the curve r is a plane curve. If the determinant dxd^y — dyd^x vanishes for some value c of the variable t between a and 6, the preceding proof fails, for the coefficients Ci and d might be infinite or indeterminate at such a point. Let us suppose for definiteness that the preceding determinant vanishes for no other value of t in the interval (a, 6), and that the analogous determinant dx(Pz — dzd^x does not vanish for t = c. The argument given above shows that all the points of the curve r which correspond to values of t between a and c lie in a plane P, and that all the points of r which correspond to values of t between c and b also lie in some plane Q. But dxd^z —dzd'^x does not vanish for t = c; hence a number h can be found such that that minor does not vanish anywhere in the interval (c — A, c ■{■ h). Hence all the points on T which correspond to values of t between c — h and c -\- h must lie in some plane R. Since R must have an infinite number of points in common with P and also with Q, it follows that these three planes must coincide. Similar reasoning shows that all the points of r lie in the same plane unless all three of the determinants dxd^y -dyd^x, dxd^z — dzd^x, dyd^z — dzd^y vanish at the same point in the interval (a, h). If these three determinants do vanish simultaneously, it may happen that the curve V is composed of several portions which lie in different planes, the points of junction being points at which the osculating plane is indeterminate.* II all three of the preceding determinants vanish identically in a certain intefTal, the curve r is a straight line, or is composed of several portions of straight lines. If dx/dt does not vanish in the interval (a, 6), for example, we may write a^ydx -dyd^z _ ^ d^zdx -dzd^x _ , whence dy = CidXy dz = Cadx, •This singular case seems to have been noticed first by Peano. It is evidently of Interest only from a purely analytical standpoint. XI. §217] OSCULATING PLAHK 4^7 where Ci and Ct are oomtiiili. FbuOly, uoUmt »-1^trmim 0n§ which shows that r is a straight line. S17. Stationary taag«ato. The pireoeding parifraph MgpaU tJ certain points on a skew curve whieh we had not pi ti kMM|y the points at which we hate (11) *£ = *^ = *f. ^ ' dx dy dM The tangent at such a point la eald to be iiatianarjf. it u eaaj to i formula for the distance between a point and a ■M^fh< Una UmI Ite from a point of r to the tangent at a neighboring point, whIeh it infinitesimal of the second order, ia of the third order for a If the given curve F is a plane cunref the itatiftnary Ungenti are tbt Unfiliel the poinu of inflection. The preceding paragraph ihowi that tha only eortt whose tangents are all stationary ia the itraight Una. At a point where the ungent is stationary, A = 0, and the aqnatloa ol Iht osculating plane becomes indeterminate. But in general thia ladolemlaatkM can be removed. For, returning to the calcnlation at the heglnnlag of | SIS and carrying the expansions of the oottrdinaiee of Jf ' to terms of tho third crter, it is easy to show, by means of (11), that the equation of the pliM thro«g|l JT and the tangent at 3f is of the form X-x Y-y Z-t nt) ^'(0 f(o 0. where ci , cj , e* approach zero with k. Hence that plane appitMMM a ; definite limiting position, and the equation of the oa cn l ating plane li replacing the second of equations (0) by the aqoadon ^d»z + Bd»y + Cd"« = 0. If the coordinates of the point M also satisfy ths sqantlOB d»g _ d^W _d^* dx ^ dy " dx' the second of the equations (6) should be replsosd by ths sqnattos ^dfx = B<itif-f CdrsKO, where q is the least integer for which this lattsr eqnatkm is dlMlMC tnm ths equation Adx= Bdy + Cdx t= 0. Ths proof of this sti nation of the behavior of the curre with rs^MOt to its os rohtin g plaas sfs Ml to the reader. Usually the preceding equaUon Inrolring the third diflerentials is «AflisM» and the coefficienU ^, B, C do not satisfjr ths s^oslkm ii d>s •»- Bdhy + C#t ai 0. In this case the curre crosses s?sry tangent phuis sxespt ths 458 SKEW CURVES [XI, §218 218. Special earvet. Let us consider the skew curres F which satisfy a relation of the form (12) xdy -ydx = Kdz, where IT is a given constant. From (12) we find immediately ( xcPy- yd^x = KcPz, ***' \xd*y-yd»x + dzd^y-dyd^x = Kd^z. Let us try to find the osculating plane of T which passes through a given point (a, 6, c) of space. The coordinates (x, y, z) of the point of tangency must satisfy the equation a — X b — y c — z dx dy dz = 0, d^x d^y d^z which, by means of (12) and (13), may be written in the form (14) ay-bx-\-K(c-z) = 0. Hence the possible points of tangency are the points of intersection of the curve r with the plane (14), which passes through (a, 6, c). Again, replacing dz, d^z and d^z by their values from (12) and (13), the equa- tion A = 0, which gives the points at which the osculating plane is stjationary, becomes A = ^{dxd^y -dyd^xY = 0; hence we shall have at the same points d^x _ d2y ^ yd^x — xd'^y _ d^ dx dy ydx — xdy dz which shows that the tangent is stationary at any point at which the osculating plane is stationary. It is easy to write down the equations of skew curves which satisfy (12) ; for example, the curves x = At^, y=:Bt^, z = Ct«' + «, where A, B, C, m, and n are any constants, are of that kind. Of these the simplest are the skew cubic x = t, y = t^j z = t', and the skew quartic x = t, y = t^,z = t^. The circular helix 05 = a cost, y = aamt, z = Kt is another example of the same kind. In order to find all the curves which satisfy (12), let us write that equation in the form d{xy - Kz) = 2ydx. If we Mt «=/(t), xy-Kz = <f>{t), the preceding equation becomes 2y/'(0 = ^'(0. XI, §219] ENVELOPES OF .mk^al^.jj 469 Solving these three equatioiui (or x, y, and s, w iad tiM fMMil tqmi^imm «C P in the form where /(t) and ^{t) are arbitrary fnnctiona of the paraaMler t It la d«r, llov- ever, that one of these fanctloos may be iiffnml at laadom wIthoiH iam of generality. In fact we may set /(f) = f, daoe this aaoanli to drnMlac/tr) aa a new parameter. n. ENVELOPES OF SURFACES Before taking up the study of the ounrature of ikew ennret, w% shall discuss the theory of enyelopes of surfaoea. 219. One-parameter families. Let 5 be a surface of the family (16) /[x,y,«,a) = 0, where a is the variable parameter. If there exiata a surfaoe B which is tangent to each of the surfaces S along a curve C, the torfiee S is called the envelope of the family (16), and the curre of C of the two surfaces S and E is called the eharaeUritiie In order to see whether an envelope exists it is evidently sary to discover whether it is possible to find a curve C oo eaeh of the surfaces S such that the locus of all these carves is taafeni to each surface S along the corresponding curve C Let (x, y, u) be the coordinates of a point A/ on a oharacteristio. If 1/ is ooi a singular point of S^ the equation of the tangent plane to 5 al if it As we pass from point to point of the surfaoe i?, «, jf , c, and a are evidently functions of the two independent variables whieh express the position of the point upon K, and theee functions satisfy the equation (16). Hence their differentials satisfy the relation Moreover the necessary and sufficient oonditioik that the plane to E should coincide with the taogeol plane lo S is or, by (17), (18) g-0. 460 SKEW CURVES [XI, §220 Conrepsely, it is easy to show, as we did for plane curves (§ 201), that the equation R(xj y, z) = 0, found by eliminating the param- eter a between the two equations (16) and (18), represents one or more analytically distinct surfaces, each of which is an envelope of the surfaces S or else the locus of singular points of 5, or a com- bination of the two. Finally, as in § 201, the characteristic curve represented by the equations (16) and (18) for any given value of a is the limiting position of the curve of intersection of S with a neighboring surface of the same family. 220. Two-parameter families. Let S be any surface of the two- parameter family (19) f(x,y,z,a,b)=0, where a and b are the variable parameters. There does not exist, in general, any one surface which is tangent to each member of this family all along a curve. Indeed, let b = <l>(a) be any arbitrarily assigned relation between a and b which reduces the family (19) to a one-parameter family. Then the equation (19), the equation b = ^(a), and the equation represent the envelope of this one-parameter family, or, for any fixed value of a, they represent the characteristic on the correspond- ing surface S. This characteristic depends, in general, on '<t>'(a), and there are an infinite number of characteristics on each of the surfaces S corresponding to various assignments of <^(a). There- fore the totality of all the characteristics, as a and b both vary arbi- trarily, does not, in general, form a surface. We shall now try to discover whether there is a surface E which touches each of the family (19) in one or more points, — not along a curve. If such a surface exists, the coordinates («, y, z) of the point of tangency of any surface S with this envelope E are functions of the two variable parameters a and b which satisfy the equation (19) ; hence their dif- ferentials rfx, rfy, dz with respect to the independent variables a and b satisfy the relation XI. §221] SNVELOPBS OF SURPACES 4«1 Moreover, in order that the surface which it Um loeui of tiM pofail of tangency (or, y, z) should be tan^oat to 5, U it alto innmiij that we should have or, by (21), Since a and b are independent variables, it follows that the equat¥nit must be satisfied simultaneously by the coordinates (as, y, u) of tiM point of tangency. Hence we shall obtain the eqiuKion of tho envelope, if one exists, by eliminating a and b between the thftt equations (19) and (22). The surface obtained will sorely be tan- gent to ^' at (x, y^ z) unless the equationa dx dy d* are satisfied simultaneously by the values (x, y, «) whieh tatiafjr (19) and (22) ; hence this surface is either the envelope or else the locos of singular points of S. We have seen that there are two kinds of envelopes, **tt**^ on the number of parameters in the given family. For example, the tangent planes to a sphere form a two-paramotar family, and each plane of the family touches the surface at only one point On the other hand, the tangent planes to a oone or to a eylinder form a one-parameter family, and each member of the lunily it tangent to the surface along the whole length of a 221. Developable surfaces. The envelope of any one-parfttar JMnily of planes is called a developabU tur/ace. Let (23) « = aar + y/(a) + 4(a) be the equation of a variable plane P, whtre a is a ^liimtlM nod where /(a) and 4>(a) are any two fonetions of a, Tbtn Ihn 9qu^ tion (23) and the equation (24) « + yA«) + ♦'(«)-<> repi-esent the envelope of the family, or, for a givtn Tihitof «, Ibtj represent the characteristic on the ooRttpooding plants All Ibitt 462 SKEW CURVES [XI, §221 two equations represent a straight line; hence each characteristic is a straight line G, and the developable surface is a ruled surface. We proceed to show that all the straight lines G are tangent to the same skew curve. In order to do so let us differentiate (24) again with regard to a. The equation obtained (26) y/"(«) + ^"(a) = determines a particular point M on G. We proceed to show that G is tangent at M to the skew curve V which M describes as a varies. The equations of T are precisely (23), (24), (25), from which, if we desired, we might find jc, y, and z as functions of the variable parameter a. Differentiating the first two of these and using the third of them, we find the relations (26) dz = adx -\-f{a)dy, dx +f{a) dy = 0, which show that the tangent to T is parallel to G, But these two straight lines also have a common point ; hence they coincide. The osculating plane to the curve r is the plane P itself. To prove this it is only necessary to show that the first and second differentials of a, y, and z with respect to a satisfy the relations dz = adx +f(cc) dy, d:^z = ad^x-{-f(a)d^y. The first of these is the first of equations (26), which is known to hold. Differentiating it again with respect to or, we find d^z = ad^x ^f(a)d^y + [dx +f(a)dy^da, which, by the second of equations (26), reduces to the second of the equations to be proved. It follows that any developable surface may he defined as the locus of the tangents to a certain skew curve T. In exceptional cases the curve r may reduce to a point at a finite or at an infinite distance ; then the surface is either a cone or a cylinder. This will happen whenever f"(a) = 0. Conversely, the locus of the tangents to any skew curve r is a developable surface. For, let be the equations of any skew curve T. The osculating planes A(X - x) + B(Y -y)-hC(Z-z) = XI. §221] ENVELOPES OF SURFACES form a ono-parameter lamilj, whose eDT«lop« it giv«o bj Um pi» cediug equation and the equation rf^ ( X - «) + rf5( r - y) + dC(Z ^ .) - . For any fixed value of t the same equations represent the ebarse- teristic in the corresponding osculating plane. We shall show thai tliis characteristic is precisely the tangent at the point of r. It will be sufBcient to establish the eqnatioos Adx + Bdy -^Cdz = 0, dA dx -^ dBdy + dC dM » 0. The first of these is the first of (6), while the seoood is sasily obtaineil by differentiating the first and then tw^inj qj^ q| ||^ second of (6). It follows that the characteristio is parallel to the tangent, and it is evident that each of them peisos thioofh the point (x, y, z)\ hence they coincide. This method of forming the developable gives a elear idea of the appearance of the surface. Let .4 B be an arc of a skew curve. At each point M of AB draw the tangent, and cooaider only that half of the tangent which extends in a certain direolkNi, — frooi A toward B, for example. These half rays form one nappe 8i of the developable, bounded on three sides by the arc AB and the taA> gents A and B and extending to infinity. The other ends of the taa- gents form another nappe .S', similar to S^ and joined to S, aloof Ibo arc A B. To an observer placed above them thrae two nappes nppeor to cover each other partially. It is evident that any plane not tan- gent to r through any point O oi AB cuts the two nappes ^*i and 5, of the developable in two branches of a curve which has a cusp at O, The skew curve T is often called the tdge ^ r t f rt §t Um of the developable surface.* It is easy to verify directly the statement just made. Let us take O as origin, the secant plane as the ary plane, the tangent to T as the axis of z, and the osculating plane as the xs plana. Aanmiaf that the coordinates x and y of a point of V can be expanded b poweie of the independent variable «, the equations of T are of the form a: = a,«* + at«*+ yo *.«•+•••, for the equations dM dM d^ • The English term " edge of l egiiM l oi i " does MS ■>!■■* •^ ^ T'^^^ of cusps. The French terme^srite del are more suggestive. — Trajh. 464 SKEW CURVES [x; §222 must be satisfied at the origin. Hence the equations of a tangent at a point near the origin are 2arz + '" ^b^z^■\-"' Setting Z = 0, the coordinates X and Y of the point where the tan- gent meets the secant plane are found to have developments which begin with terms in z^ and in «*, respectively ; hence there is surely a cusp at the origin. Example. Let us select as the edge of regression the skew cubic « = <, y = <« , z = t*. The equation of the osculating plane to the curve is (27) t«-3t2X+3<r-Z = 0; hence we shall obtain the equation of the corresponding developable by writing down the condition that (27) should have a double root in t, which amounts to eliminating t between the equations t2_2<X + r=0, Xt^-2tY + Z = 0. (28) I The result of this elimination is the equation (XF - z)2 - 4(X2 - r)(r2 - xz) = o, which shows that the developable is of the fourth order. It should be noticed that the equations (28) represent the tangent to the given cubic. 222. Differential equation of developable surfaces. If « = F(a;, y) be the equation of a developable surface, the function F(x, y) satisfies the equation s^ — rt = 0, where r, s, and t represent, as usual, the three second partial derivatives of the function F(x, y). For the tangent planes to the given surface, Z =:pX + qY + z-px-qyy muet form a one-parameter family ; hence only one of the three coefficients /?, q^ and z—px — qy can vary arbitrarily. In particular there must be a relation between p and q of the form f{p, q) = 0. It follows that the Jacobian D{py q)/D(x, y) = rt — s^ must vanish identically. Conversely, if F(x, y) satisfies the equation rt — s^ = 0, p and q are connected by at least one relation. If there were two distinct relations, p and q would be constants, F(x, y) would be of the form ax-^by -^-Cy and the surface z = F(ic, y) would be a plane. If there XI, 1228] ENVELOPES OF MiitACES 466 is a single relation between p and y, it may be writtm in the form 7 —f(p)f ^^^re p do69 not reduce to a eonstani. But we also hm9% hence z — px — qy is also a function of p, lay f(p), wb«Mf«r rt — 8^ z=0. Then the unknown function F(x, y) and it« ptH ial derivatives p and y satisfy the two equations y = <i>(P)f « -;w - ^rty - f(p). Differentiating the second of these equations with retpect to x aod with respect to y, we find [^ + y <I>'(P) + 'A'C/')] ^ = 0, [x + y ^'Cp) -»- ^'(rt] ^ - 0. Since p does not reduce to a constant, we must baro hence the equation of the surface is to be found by alinuDittiig • between this equation and the equation «=iw; + y^p)-f f(/)), which is exactly the process for finding Uie envelope of the lanjily of planes represented by the latter equation, /> being thought of as the variable parameter. 223. Envelope of a family of skew aurea. A one-paramcUr fiMily of skew curves has, in general, no envelope. Let us onniidf finl a family of straight lines (29) X = «»+/», y = *«+ • where a, b^ p^ and q are given functionii of a variable piiimtw «. We shall proceed to find the conditions undor whifih orstj m a«bs t of this family is tangent to the same skew conre T. Let « « 4(a) be the z coordinate of the point ^1/ at which the rariable stvmifht line D touches its envelope T. Then the roqaired curve T will be represented by the equations (29) together with the eqnaiioo z = <^(a), and the dire<^tion cosines of the tangvot lo T will be pto* portional to dx/da, dyjda^ d*/da, i.e. to the three qotntitiee a <^'(a) -f a'«^(a) -k- p\ ft^V) + *'♦(«> + f*. 4'(a\ 466 SKEW CURVES [XI, § 223 where a', b', p\ and q' are the derivatives of a, h, p^ and q, respec- tively. The necessary and sufficient condition that this tangent be the straight line D itself is that we should have dx _ dz da da dy _ .d& da da that is, a'4>{a) +1?' = 0, h'^{a) + g'' = 0. The unknown function 4>{a) must satisfy these two equations; hence the family of straight lines has no envelope unless the two are compatible, that is, unless a'q^ - Vp' = 0. If this condition is satisfied, we shall obtain the envelope by setting <l>(a)=-p'/a'=-q'/b'. It is easy to generalize the preceding argument. Let us consider a one-parameter family of skew curves (C) represented by the equations (30) F(x, y, «, a) = , ^(x, 2/, «, a) = 0, where a is the variable parameter. If each of these curves C is tangent to the same curve r, the coordinates (x, y, z) of the point M at which the envelope touches the curve C which corresponds to the parameter value a are functions of a which satisfy (30) and which also satisfy another relation distinct from those two. Let dx^ dy^ dz be the differentials with respect to a displacement of M along C ; since a is constant along C, these differentials must satisfy the two equations (31) (dF^ dF . dF. _ -^ dx -\- T- dy -\- -z- dz = 0, ox Cy '' cz -^ dx -{- 1^ dy -{- ^r- dz = . ^ox ^y ^« On the other hand, let hx^ hy, Bz, Sa be the differentials of x, y, «, and a with respect to a displacement of M along r. These differen- tials satisfy the equations (32) ' dF dF dF dF d^ dz ye.^^'P^^'i^^Tj-' XI, §223] £NVKLOP£S OF SURFACES 467 The necessary and sufficient conditions that the earrm C tad r be tangent are dx _dy dm ix " iy " $g* or, making use of (31) and (32), It follows that the coordinaiet (x, y, s) of the point o/temgmt^ Miuf satisfy the equations (33) F=0, ♦ = 0, |f = 0. g-0. Hence, if the family (30) is to have an envelope, the four equattoos (33) must be compatible for all values of a. Conversely, if tbcM four equations have a common solution in a*, y, and c for all Taloee of a, the argument shows that the curve T described by the point (ar, y, z) is tangent at each point (x, y, x) upon it to the ing curve C. This is all under the supposition that the ratios dx^ dy, and dz are determined by the equations (31), that it, that tha point (a;, y, z) is not a singular point of the curve C. Note. If the curves C are the characteristics of a ODe-paramsIrr family of surfaces F(x, y, ;:;, a) = 0, the equations (33) reduoe to the three distinct equations dF ^F (34) F=0, —=-0, V^ = 0: hence the curve represented by these equat ♦' . .• •' :■•• of the characteristics. This is the generali/u • ■ : i • i: - ' m proved above for the generators of a developable sorteee. The equations of a one-parameter family of straight in the form (36) _,___-,__. where Xo, Vq> ^o^ ^^ ^, <" arc itincuonB of a variable paramelar a. U ls< find directly the condition that thb family iboold have an denote the common value of each of the preoedlof imtios ; of any point of the straight line are given by the eqaslloiis and the question is to determine whether it Is posrfhit to Mhrtllals lor I nea a function of a that the variable straight line ahoald alwara remain tancaal lo 468 SKEW CURVES [XI, § 224 the curve described by the pomt (x, y, z). The necessary condition for this is that we should have xii-^a'l ^ yh-^l/l ^ zf> + Cl a b c (36) Denoting by m the common value of these ratios and eliminating I and m from the three linear equations obtained, we find the equation of condition («n x6 yo ^ a h c or h' c' = 0. If this condition Lb satisfied, the equations (36) determine {, and hence also the equation of the envelope. III. CURVATURE AND TORSION OF SKEW CURVES 224. Spherical indicatrix. Let us adopt upon a given skew curve F a definite sense of motion, and let s be the length of the arc AM measured from some fixed point A as origin to any point M, affixing the sign + or the sign — according as the direction from A toward M is the direction adopted or the opposite direction. Let MT be the positive direction of the tangent at M, that is, that which cor- responds to increasing values of the arc. If through any point in space lines be drawn parallel to these half rays, a cone S is formed which is called the directing cone of the developable surface formed by the tangents to F. Let us draw a sphere of unit radius about O as center, and let 2 be the line of intersection of this sphere with the directing cone. The curve % is called the spherical indicatrix FiQ. 48 of the curve F. The correspondence between the points of these two curves is one-to-one ; to a point Af of F corresponds the point m where the parallel to A/r pierces the sphere. As the point Af describes the Xl.$22fl] CURVATURE TORSION 4$% curve r in the positive sense, the point m deecrtbee the eurre 1 ui a certain sense, which we shall adopt as poeitiTt. Then IIm ooiit sponding arcs « and <r inorease simultaaeously (Rig. 48). It is evident that if the point O be displaoed, the whoU eorre S undergoes the same translation ; hence we may suppoee tlMl O lies at the origin of coordinates. Likewise, if the positive mom od tke curve r be reversed, the curve 2 is replaced bj a curve syiBaMtrkal to it with respect to the point O \ but it should be notioed thai ibe positive sense of the tangent fiU to S is independent of tba mbm of motion on T. The tangent plane to the directing cone along the geoeialor Om it parallel to the osculating plans at M, For, let AX -^ BY -i- CZ b be the equation of the plane Omm'f the center O of the spheM at the origin. This plane is parallel to the two tangents at if at M' ; hence, if ^ and ^ + A are the parameter valuM which spond to M and M\ respectively, we most have (38) Af(t) + J5^'(0 -f Cf (0 = 0. (39) Af'(t 4- A) + Bi^Xt + A) + C^'(t + A) - 0. The second of these equations may be replaced by the equafcico , f(t + h) -fit) ^ ^ »'« + h)- »'(o , ^ »'(> -f A) - »'(» , a^ AAA which becomes, in the limit as A approaohM Mro, (40) Af\t) + B^'\t) + C^'XO - 0. The equations (38) and (40), which determine i4, B, and C for the tangent plane at m, are exactly the same as the equattonc (6) which determine A, 5, and C for the osculating plane. 225. Radius of curvature. Let • be the angle beiWMO the positive directions of the tangenU MT and itV at two neighboring poialc M and 3/' of T, Then the limit of the ratio •/arc J/JT, m M approaches ilf, is called the eurvahtre of T at the point If. just as for a plane curve. The reciprocal of the curvature is cdM Ika radius of curvature : it is the limit of arc J/JT/ti. Again, the radius of curvature R may be defined to be the limit of the ratio of the two infinitesimal arM MMt and mm\ for we hava arc AfAf' ^ arc JfiT' ^ aictam* ^ thotdwtm' M arc mm' chord 470 SKEW CURVES [XI, §225 and each of the fractions (arc mm ')/(chord) mm^ and (chord mm')/i3i approaches the limit unity as m approaches m'. The arcs5(=J/M') and a-(=mm') increase or decrease simultaneously; hence ds («) ^ = 1-. Let the equations of r be given in the fotm (42) x==f(t), y = <t>(t), z = ^{t), where O is the origin of coordinates. Then the coordinates of the point m are nothing else than the direction cosines of MT, namely _ ^^ Q _ ^y — ^^ ds ds ^ ds Differentiating these equations, we find dsd^x —dxd^s .^ dsd^y — dyd^s , dsd^z — dzd^s ""= — dT^ — ' '^^^ ds' ' "y^ — s^ d.' = d<^ + d^ + dr' = S{ds<^x-dxdH)\ where O indicates as usual the sum of the three similar terms obtained by replacing x by a;, y, z successively. Finally, expanding and making use of the expressions for ds^ and ds d^Sj we find _ Sdx^ S{d^xf-\Sdxd^xT ds* By Lagrange's identity (§ 131) this equation may be written in the form dxd^Zj a notation which we shall use consistently in what follows. Then the formula (41) for the radius of curvature becomes (44) R*= "^'^ da* = y ds* where (43) r = dyd^z-dzd^y, B = dz d^x C = dxd^y — dyd^Xj and it is evident that R* is a rational function of x, y, z, x\ y\ z\ «", y", «". Tlie expression for the radius of curvature itself is irrational, but it is essentially a positive quantity. XI, §226] CURVATURE TORhlUN 471 Note. If the independent Tariable aeleeted ia the tra « of Iho curve r, the functions /(«), ^(«), and f («) ntiify the eqaatioo Then we shall have (46) " t/a =/"(«) (if, <//3 = ^"Wrfi, dy . f (•) A. and the expression for the radius of ounrature iMumet tha pMtae- ularly elegant form (44') ;^. = [/"(')]• + [♦"(')]* + [*"(»)?• Principal normal Center of curratnre. Let us draw a line through 3/ (on F) parallel to mt, the tangent to 1 at m. Let MN be the direction on this line which corresponds to the positlTe diree> tion vit. The new line MN is called the principal normal to T at i# : it is that normal which lies in the osculating plane, sinee wU is perpendicular to Om and OnU is parallel to the oecolatinf piMM (§ 224). The direction MN is called the pomUm dirmiiam ^ Os principal normal. This direction is uniquely defined, since the posi- tive direction of mt does not depend upon the choice of the positive direction upon F. We shall see in a moment how the direetta hi question might be defined without using the indicatiiju If a length MC equal to the radius of corvatore at 1# be laid off on iMX from the point 3/, the extremity C is called the curvature of F at A/, and the circle drawn around C in the ing plane with a radius MC is called the rireis of emrvoHtr*. a\ p, y be the direction cosines of the principal nonnaL coordinates (xi, yi, «i) of the center of curvature are arj = X -f Ra\ yi = y + Rff* »i - » + Ry*' But we also have , da dads ^da , dsd^m - datd^e da d* da d* d^ and similar formulae for ff and y'. Replacing a' by iU value in the expression for x, we find .dMd^x ^djtd^s 472 SKEW CURVES [XI, §226 But the coefficient of R^ may be written in the form ds* "" ds* or, in terms of the quantities A, B, and C, Bdz — Cdy ds^ The values of yi and z^ may be written down by cyclic permutation from this value of Xi , and the coordinates of the center of curvature may be written in the form '' , ^^ Bdz — Cdy x^ = x + R'' —. ^j (46) ds"" h C dx — A dz 2" = 2' + « 1? ' These expressions for x^ yi, and z^ are rational in x, y, z, x\ y\ z\ x", 2/", «". A plane Q through M perpendicular to MN passes through the tangent MT and does not cross the curve r at M. We shall proceed to show that the center of curvature and the points of V near M lie on the same side of Q. To show this, let us take as the independent variable the arc s of the curve r counted from M as origin. Then the coordinates X, Y, Z oi a point M ' of F near M are of the form x^- s dx s^ /d^x \ 1 ^ "^ 172 V"^ 7 the expansions for Y and Z being similar to the expansion for X But since s is the independent variable, we shall have dx d^x da da da- 1 , 57="' ds^ ~ ds ~ da- ds ~ R and the formula for .Y l)ecomes ^=^ + - + (l'+')A If in the equation of the plane Q, aXX - ar) 4- p'( V - y) + y\Z - ;^) = 0, XI, §227] CUBVATLKE T0R810N 47t Xf Y, and Z be replaced by these expaoaiona in the left-hand wanibM the value of that member ia found to be where 17 approaches zero with «. Thia quantity ia poeittTe for all values of s near zero. Likewiae, replacing (Jt, K, Z) by the eoSidi- nates (x + Jia'j y + Rp^ z + Ry^ of the center of canntui% the result of the substitution is R, which ia eaaentially poeitive. Heaee the theorem is proved. 227. Polar line. Polar surface. The perpendicular A to the oeeu- lating plane at the center of curvature ia called the polar iims. Thia straight line is the characteristic of the normal plane to P. For, in the first place, it is evident tliat the line of iiitnuectiou D oi the normal planes at two neighboring pointa M and M* ia perpeodienlar to each of the lines MT and M'T' ; hence it is also perpendicular to the plane ynOm'. As M' approaches M, the plane mOm* approaches parallelism to the osculating plane ; hence the line D approtehes a line perpendicular to the osculating plane. On the other hand, to show that it passes through the center of curratorey let « be the independent variable ; then the equation of the normal plane is (47) a(X - x) + ^(r- y) 4- y(Z - a) = 0, and the characteristic is defined by (47) together with the equation (48) ^(.Y -x)-f|V-y) + ^'(^ -«)-!- 0- This new equation represents a plane perpendicular to the principal normal through the center of curvature; hence the intsneetkn of the two planes is the polar line. The polar lines form a ruled surface, which ia called the ^slnr surface. It is evident that this surface ia a developable, atnee we have just seen that it is the envelope of the normal plane to T. If r is a plane curve, the polar surface ia a oylinder whoea right section is the evolute of T ; in this special case the preceding ments are self-evident. 228. Torsion. If the worda "tangent line" in the daOBitkn of curvature (§ 225) be replaced by the words «oeeulatinf ptsB^" » new geometrical concept ia introduced which mmtnim, in a the rate at which the osculating plane tuma. IM «' be the angle between the osculating planea at two neighboring poinU J/ and if'; 474 SKEW CURVES [XI, §228 then the limit of the ratio <o'/arc MM', as M approaches M', is called the torsion of the curve r at the point M. The reciprocal of the torsion is called the radius of torsion. The perpendicular to the osculating plane at M is called the binormal. Let us choose a certain direction on it as positive, — we shall determine later which we shall take, — and let a", j8", y" be the corresponding direction cosines. The parallel line through the origin pierces the unit sphere at a point n, which we shall now put into correspondence with the point M of r. The locus of w is a spherical curve 0, and it is easy to show, as above, that the radius of torsion T may be defined as the limit of the ratio of the two corre- sponding arcs MM' and nn' of the two curves F and ®. Hence we shall have d dr" where t denotes the arc of the curve ©. The coordinates of n are a", )8", y", which are given by the formulae (§ 215) a"=— ^=4_, p"= , ^ =:> y' = , ^ ±Va^+¥+c^ ±\/a^+b^-\-c^ ±V^2+i52-fc« where the radical is to be taken with the same sign in all three formulae. From these formulae it is easy to deduce the values of da"y dp'\ rfy"; for example, ^^„ ^ . (A' -\-B^ + C^dA-A{A dA -hBdB-^CdC) (A^ + B^-\- C^)^ whence, since dr^ = da"^ + dp"^ -f dy"% ,^.^ SA^SdA^-[S(AdA)r ^ (A^ + B^ -{- Cy or, by Lagrange's identity, SiBdC-CdBf {A^ + B^-^ Cy • where o denotes the sum of the three terms obtained by cyclic per- mutation of the three letters A, B,C. The numerator of this expres- sion may be simplified by means of the relations Adx+ Bdy+ Cdz = 0, dA dx -f dBdi/ + dCdz== 0, whence 1 ^'"^ BdC-CdB CdA^AdC AdB-BdA -F ^^ 5"«j CLHVATURE TORSION 475 where /C is a quantity deBned by the equatioii (49) itatll This gitct where /T is defined by (49) ; or, expanding, = S{dzd*xd*ij-~dxcPMd*y), where aS* denotes the sum of the three terms obtained by oyelie per- mutation of the three letters x, y, z. But this value of IT it cxaelly the development of the determinant A [(8), S 216]; hence and therefore the radius of torsion is given by the formula (50) r=±ill±^l±^. A If we agree to consider T essentially positive, ac we did the radios of curvature, its value will be the absolute value of the teoocid OMOi* ber. But it should be noticed that the expression f or T is ratk»al in X, y, «, x'j y\ «', x", y", «"; hence it is natural to represent the radius of torsion by a length affected by a sign. The two sifM which T may have correspond to entirely different aspects of Um curve r at the point M. Since the sign of T depends only on that of A, we shall iiiTestigstt the difference in the appearance of T near M when A has different signs. Let us suppose that the trihedron Oryt is placed so thai an observer standing on the xy plane with his fsel at O and his hsnd in the positive z axis would see the x axis turn throogh W t§ kkl^ if the X axis turned round into the y axis (see fooCnole, p. 477). Suppose that the positive direction of the binormal MS^ has been so chosen that the trihedron fonned from the lines If T, MN, MS^ has the same aspect as the trihedron formed from the lines Om^Op,OB; that is, if the curve T be moved into such a position thai JT eoineides with O, i\fT with Ox, and MN with Oy, the dirsetioo MN^ will eote* cide with the positive z axis. During this motion the absolvis Talne of T remains unchanged ; hence A cannot Tsnish, and hioee it 476 SKEW CURVES [XI, § 228 even change sign.* In this position of the curve r with respect to the axes now in the figure the coordinates of a point near the origin will be given by the formulae (61) B where c, c', e" approach zero with f, provided that the parameter t is so chosen that ^ = at the origin. For with the system of axes employed we must have dy = dz = d^z = when ^ = 0. Moreover we may suppose that a^ > 0, for a change in the parameter from t to — t will change ai to — aj . The coefficient b^ is positive since y must be positive near the origin, but c^ may be either positive or negative. On the other hand, f or ^ = 0, A = 12a^h<^c^ dt^. Hence the sign of A is the sign of c^. There are then two cases to be distinguished. If c, > 0, X and z are both negative f or — A < ^ < 0, and both positive for < < < A, where A is a sufficiently small positive number ; i.e. an observer standing on the xy plane with his feet at a point P on :n N 'M' Fia. 49, a M Fig. 49, 6 \M' the positive half of the principal normal would see the arc MM^ at his left and above the osculating plane, and the arc MM" at his right below that plane (Fig. 49, a). In this case the curve is said to be sinistrorsal. On the other hand, if Cg < 0, the aspect of the curve would be exactly reversed (Fig. 49, b), and the curve would be said to be dextrorsal. These two aspects are essentially distinct. For example, if two spirals (helices) of the same pitch be drawn on the same right circular cylinder, or on two congruent cylinders, they will be superposable if they are both sinistrorsal or both dextrorsal ; but if one of them is sinistrorsal and the other dextrorsal, one of them will be superposable upon the helix symmetrical to the other one with respect to a plane of symmetry. * It would be easy to show directly that A does not change sign when we pass from one set of rectangular axes to another set which have the same aspect. XI, $229] CURVATURE TORSION 477 In consequence of these rest (52) r = - In consequence of theee retolto we ihall wrile A i.e. at a point where the curve is dextrorsal T ihall be poeitive, while T shall be negative at a point where the curve is sinistrofiiL A dif> ferent arrangement of the original ooOrdinate trihedron OmgM would lead to exactly opposite results.* 229. Frenet'8 formulfe. Each point if of T is the vertex of a tii- rectangular trihedron whose aspect is the same as that of the trib^ dron Oxi/z, and whose edges are the tangent, the principal Doraal, and the binormal. The positive direction of the principal normal is already fixed. That of the tangent may be chosen at pleasure, but this choice then fixes the positive direction on the binonnaL The dif* ferentials of the nine direction cosines (a, /J, y), (a\ /J*, y*), (a", /f , y*^ of these edges may be expressed very simply in terms of R, T, and the direction cosines themselves, by means of certain fonnuls doe to Frenet-t We have already found the formuls for da^ dfi, and dy: ^^"^^ ds R ds R ds R The direction cosines of the positive binormal (§ 228) are a"=e-;=4==, fl" = «-==J=, y*'«€ , ^ i> ^A^-hB^+c^ y/A*T^Tc* Va*V¥Tc^ where c = ± 1. Since the trihedron (MT, MN, MN^) has the «MM aspect as the trihedron Oxy«, we must have .' = rr-^r", or <'-'* ^XZc ' On the other hand, the formula for da" may be writteo „^ BjBdA -Adm-^CjCdA -AdC) (A*-^B»+C^^ or, by (49) and the relation /T = A, da" ^ C/8~By ^ g^A • It is usual in America to adopt aa anaagWBMt «f described above. Hence we •bookl write r« ->• U' + ^ + C^/A, «a. the footnote to formula (54), §2».— TaAJia. t NouvelUs Annate* de IfalAtfrnoNfUM, l»«. ^ »i> 478 SKEW CURVES [XI, §229 The coefficient of a' is precisely 1/T, by (52). The formulae for rfj3" and dy" may be calculated in like manner, and we should find (^) IT^J' -dT-f' ds -r' which are exactly analogous to (53).* In order to find da\ dp'j dy\ let us differentiate the well-known formulae aa' + )8)3' + yy' =0, a'a" + ^')8"+y'y" = 0, replacing da, dp, rfy, rfa", dff\ c^y" by their values from (53) and (54). This gives a'da' + p'dp' + y'dy =0, ds a da'+/3 dp + y ^^y' + — = 0, d<i a''da' + P"dp-\-y"dy-hj; = 0', whence, solving for da', dp, dy\ da' a a" dp p p" dy'_ _z_y:. ds R t' ds R t' ds R T (65) The formulae (53), (54), and (55) constitute Frenet's formulae. Note. The formulae (54) show that the tangent to the spherical curve described by the point n whose coordinates are a", p\ y" is parallel to the principal normal. This can be verified geometrically. Let S' be the cone whose vertex is at and whose directrix is the curve 0. The generator On is perpendicular to the plane which is tangent to the cone S along Om (§ 228). Hence S' is the polar cone to S. But this property is a reciprocal one, i.e. the generator Om of S is surely perpendicular to the plane which is tangent to S' along On. Hence the tangent mt to the curve 2, since it is perpen- dicular to each of the lines On and Om, is perpendicular to the plane mOn. For the same reason the tangent nt' to the curve © is perpendicular to the plane mOn. It follows that mt and nt' are parallel. • If we had written the formula for the torsion In the form 1/7'= A/(^« + B* + C^, Frenet's formulw would liave to be written in the form da^'/da -- a'/T, etc [Hence this would be the form if the axes are taken as usual in America. —Trans.] XI, $230] CURVATURE TORSIOH 479 230. Expansion of z, y, and s la powtrt of a. Oiren two fttselioiit R = <^(.«;, T = ^(«) of an independunt variable «, the fim of whiflh is positive, there exists a skew curve r whieh is oompUUlj ^••ntii except for its position in space, and whooe radius of curvature and radius of torsion are expressed by the given equations in tetMs of the arc s of the curve counted from some fixed point upon it A rig* orous proof of this theorem cannot be given until we have riisnisoed the theory of differential equations. Just now we shall maroly show how to find the expansions for the coordinates of a poiol on tlM required curve in powers of «, assuming that such esptatioos tziii. Let us take as axes the tangent, the principal aomial, and tho binormal at 0, the origin of arcs on T. Then we shall have /~1 \d8 J 0^ 1.2 \dsV» 1.2. S\ds»/»"' where Xy y, and z are the coordinates of a point on T, But dx ds = a d^x ds* da ds a r' whence, differentiating, In general, the repeated application of Frenet's formula where i,, Af,, P, are known functions of R, T, and their soooeetiTe derivatives with respect to #. In a similar maniior the aurositivo derivatives of y and « are to be found hf replacing («, a', O ^ (A fiy P") and (y, y', y"), respectively. But we have, at the origiBt «o = 1, ^0 = 0, yo = 0, ai = 0, /aj«= 1, yi-O. of -0, flf-O, >C-li hence the formulae (66) become I- . ^ = 1" 6R' ^ ' t* i^ dR (66') {y^ 2R'eR*ds «» 6RT 480 SKEW CURVES [xi,§23i where the terms not written down are of degree higher than three. It is understood, of course, that R, T, dR/dSj • • • are to be replaced, by their values for s = 0. These formulae enable us to calculate the principal parts of cer- tain infinitesimals. For instance, the distance from a point of the curve to the osculating plane is an infinitesimal of the third order, and its principal part is — s^/6R T. The distance from a point on the curve to the x axis, i.e. to the tangent, is of the second order, and its principal part is s^/2R (compare § 214). Again, let us cal- culate the length of an infinitesimal chord c. We find c'' = x^ + 2/» + s^ = »'-j— + •••, where the terms not written down are of degree higher than four. This equation may be written in the form which shows that the difference s — c is an infinitesimal of the third order and that its principal part is s'/24i^l In an exactly similar manner it may be shown that the shortest distance between the tangent at the origin and the tangent at a neighboring point is an infinitesimal of the third order whose prin- cipal part is s^/12RT. This theorem is due to Bouquet. 231. Involutes and evolutes. A curve Fj is called an involute of a second curve F if all the tangents to F are among the normals to Fj , and conversely, the curve F is called an evolute of Fj . It is evident that all the involutes of a given curve F lie on the developable sur- face of which F is the edge of regression, and cut the generators of the developable orthogonally. Let (x, 7/, z) be the coordinates of a point M of F, (a, yS, y) the direction cosines of the tangent MT, and I the segment MM^ between M and the point Afi where a certain involute cuts MT. Then the coordinates of Mi are xi = xi-lay yi'^y-hip, «i = « + hf whence dxi = dx -\-lda -\- adl, dyx = dy'hldp + pdl, dZi = dz + Idy -\- ydl. XI, §231] CrRVVTiMV 'OKSlOir 491 In order that the curve deM;ribd(l by i/| thould be nflnftnl to 111/ it is necessary and sufficient that a dx^ -^ fidy^ 4- T^i «I hm iJi! vaftialL i.e. that we should have adx-\- fidy -^ydM + dl-^Hada-^ fidfi + ydy)mO, which reduces to ds -^tUsaO, It foUowi thai the mvoliilii to a given skew curve F may be drawn by the tame oonftnMlMNi whUk was used for plane curves (§ 206). Let us try to find all the evolutes of a given curve F, that is, let us try to pick out a one-parameter family of uormab to the given curve according to some contin- uous law which will group these normals into a developable surface (Fig. 50). Let Z> be an evolute, <f> the angle between the normal MMi and the principal nonual MN, and I the segment MP between Ai and the projection P of the point Mi on the principal nonnaL TImb tl» coordinates (xi, y^ z{) of 3f| are X, = ar4-/a' + /<T"tan^, (67) {X, = ar + '<t' + la" tan ^, yi = y + //9' + //r'tan4, «j=« + /y' + /y"tan^, as we see by projecting the broken line MPMi upoii the thrM aiM successively. The tangent to the canre described by the point y% must be the line MM^ itself, that is, we must hare dxx ^^i ^ X, - X ■" y, - y *i - « Let k denote the common value of these ratioe ; then the e o n d Hi ea dxi = ik(xi - x) may be transformed, by inserting the Tiluee ol j^ and dxi and applying Frenet's formulie, into the form arf*(l - ;^) + a'(rf/ + /tan 4^ - «) + a"[rf(/tan^) - ^ - A/tan^] - 0. The conditions rfyi =: A: (y» - y) and rf^h - *(«t - ») W^ to earthy similar forms, which may be deduced from the preoediag by n^^Mr cing (a, a', a") by ()9, ff. /T) and (y. y', yO» wn****^/- »*»« *^ 482 SKEW CURVES [XI. §231 determinant of the nine direction cosines is equal to unity, these three equations are equivalent to the set (58) ds dl -\-l tan <ft — = kl, Ids d(l tan <^) — - = kl tan <f>. From the first of these I — R, which shows that the point P is the center of curvature and that the line PM is the polar line. It fol- lows that all the evolutes of a given skew curve T lie on the polar sur- face. In order to determine these evolutes completely it only remains to eliminate k between the last two of equations (58). Doing so and replacing Ihy R throughout, we find ds = T d<f}. Hence <^ may be found by a single quadrature : (59) * = *« +X f ' If we consider two different determinations of the angle <^ which correspond to two different values of the constant <^o> the difference between these two determinations of <^ remains constant all along r. It follows that two normals to the curve T which are tangent to two different evolutes intersect at a constant angle. Hence, if we know a single family of normals to r which form a developable surface, all other families of normals which form developable surfaces may be found by turning each member of the given family of normals through the same angl^, which is otherwise arbitrary, around its point of intersection with r. Note I. If r is a plane curve, T is infinite, and the preceding formula gives <f> = <f>Q. The evolute which corresponds to </)o = is the plane evolute studied in § 206, which is the locus of the centers of curvature of r. There are an infinite number of other evolutes, which lie on the cylinder whose right section is the ordinary evo- lute. We shall study these curves, which are called helices, in the next section. This is the only case in which the locus of the cen- ters of curvature is an evolute. In order that (59) should be satis- fied by taking </> = 0, it is necessary that 7' should be infinite or that A should vanish identically ; hence the curve is in any case a plane curve (§216). XI, §232] CURVATURE T0B8I0H Ut Note II. If the ourre Z> is an e volute of r, it follows UmI F it ab involute of D. Honoe </«i-rf(jirAr,), where «i denotes the length of the arc of the orolute eomltd f rf some iixed point. This shows that all the erolntM of utf fivw curve are rectiiiable. 288. Helic«t. Let C be any plane curve and lei us la j off oa Um ular to the plane of C erected at any point m on C a InfUi aiif the length of the arc o- of C counted from aome find point A. l^ea iIm ihsv curve r described by the point M ia called a kdiz. Lai « uIm ibt pteae «C C aa the xy plane and let be the coordinates of a point m of C In terns of the are v. Thm Iht ooftjl nates of the corresponding point Jf of the onnre T wHl be m »=/(<') . y = ^(<r). t = Jr<r, where K is the given factor of proportionality. The f unciloas / and # «Urfy the relation /'« -f 0'« = 1 ; hence, from (60), dS« = (/'« + ^'« + jr«)da« = (1 + Jr«)d«r*. where a denotes the length of the arc of r. It follow* that • a v VTTT* 4 If* or, if s and o* be counted from the same point ^ on C, « s 4r Vi-F Jn,siaesH » t^ The direction cosines of the tangent to r are (61) a=-^m., p^4^. -r.-T^^- Since 7 is independent of cr, it is evident that the tangent to T angle with the z axis ; this property ia charaoterkak : Jny makes a constant angle toith afixtd ttraigkt Urn ia a Aelic In order to prvf* this, let us take the z axis parallel to the given aumlgbt Una, and lai C be iha projection of the given curve T on the cy plane. TIm eqvatloaa of T aai be written in the form (62) «=/(<r), |f = f(«r). m^H^^ where the functions / and ^ satisfy the relatloB /'• + #'» = 1, for tWs amounts to taking the arc cr of C as the ludspaidsni varlabla. It hence the necessary and sufBcient condition thai 7 ka eoMtiat Is ihaif' be constant, that is, that ^(<r) should be of the ior« Xr + s». I* ' " the equations of the curve V will be of the fona (•©) If iba the point i = 0, y = 0, t = to- Since 7 is consuni, the formuU dy/da « Y/M sbowt th atV _ principal normal is perpendicular to the fsiisnlnfs ef the QrttaAsr. Steea k Is also perpendicular to the tangent to the balls, H la 1 therefore the osculating plane is nonnal to the ejl 484 SKEW CURVES [xi,§232 binormal lies in the tangent plane at right angles to the tangent to the helix ; hence it also makes a constant angle with the z axis, i.e. 7" is constant. Since 7' = 0, the formula d77d« = - 7/B - 7"/ T shows that 7/JB + Y' / T = 0; hence the ratio T/B. is constant for the helix. Each of the properties mentioned above is characteristic for the helix. Let us show, for example, that eo&ry curve for which the ratio T/B is constant is a heliz. (J. Bertrand.) From Frenet's formulae we have da _dp_ _ dy_ _ ^ _ J_ , da'' ~ d/3" ~ d7" ~ B~ H* hence, if ^ is a constant, a single integration gives a" = na-A, ^' = n^-B, 7" = H7 - C, where A, By C are three new constants. Adding these three equations after multiplying them by a, /S, 7, respectively, we find Aa -\- Bp -h Cy = fl", or Aa -\- Bp -\- Cy H V^2 + 52 4. (72 V^2 + 52 _|. era But the three quantities A B are the direction cosines of a certain straight line A, and the preceding equa- tion shows that the tangent makes a constant angle with this line. Hence the given curve is a helix. Again, let us find the radius of curvature. By (53) and (61) we have whence, since 7' = 0, This shows that the ratio (1 + K^/B is independent of K. But when ^ = this ratio reduces to the reciprocal 1/r of the radius of curvature of the right section C, which is easily verified (§ 205). Hence the preceding formula may be written in the form B = r{l + K^)^ which shows that the ratio of the radius of curvature of a helix to the radius of curvature of the corresponding curve C is a constant. It is now easy to find all the curves for which B and T are both constant. For, since the ratio T/B is constant, all the curves must be helices, by Bertrand's theorem. Moreover, since jB is a constant, the radius of curvature r of the curve C also is a constant. Hence C is a circle, and the required curve is a helix which lies on a circular cylinder. This proposition is due to Puiseux.* • It is assumed in this proof that we are dealing only with real curves, for we asBumed that A^ + li^ -^ C* does not vanish. (See the thesis by Lyon : Sur les courbea a torsion constante, 1890.) XI, f 233] CURVATURE TORSION S88. B«rtnnd'a cottm. Hm prfaieipal principal noriualii to an Inflaita munbtr ci given curre. J. Beitnod tttooiptod to iad in curyes whose principal nonnali are the prineipal ■ofik lo » 099m dbtm ourre r. Let the ooOrdinatei x, y, s of a point of r be givea ae fttailaM el ifca arc f. Let ua lay off on each prindpal nonnal a eegaaat of \m^h t, wmi hi tt* coordinates of the extremity of this wginwn be X, F, 2 ; thm w iMI hmt (64) X = x + ta', T^w-^lfiT, ZmM^kf, The necessary and sufficient condition that the priDeipal nnr— I lollMcwf* r* described by Uie point (JT, F, Z) ibould eolndde with the prtBdyid awal to r is that the two equations o'idYd^z - dZdi^T) + /r (dz<««x - dzd^z) + v(dx#r - tfr#X) • # should be satisfied simultaneously. Tha nuanhif of aaoli of tiMM oqpMlioM li evident. From the first, di = ; hence the lengtli of tiM eag— tTibo^M hi a constant. Replacing dJT, (i*X, dY, • • • In the aeoood oqaatkm by tMr from Frenet*s formulsB and from the fonnoUs obtataad by Frenet's, and then simplifying, we finally find M(-i)-(- !)'(»• whence, integrating, (66) i"*"?"*' where V is the constant of intagratloo. It foUowa thai CJU rsgnirvrf c those for which there entU a linear rtUiUon betm$m tk§ tmn tl hn md $k On the other hand, it is easy to ahow that thia eoa dit i w i la adMnft the length I is given by the relation (66). A remarkable particular case had already baas ooHod by Italfi that in which the radius of currature Is a conaianu In that eoaa (li| I = R, and the curve r' defined by the eqoatlooa (64) la tba b»M of tba of curvature of r. From (64), ■winrtng ImB dx=^^a^^d., dr=^?,frdM. ^ — *y which show that the tangent to r la Iha polar Una of T. Tba ture ii' of r' is given by the formula ^ d x*.f<r»-i><p _-. hence /T ah» la constant aad equal to R. Tba relauon ^ tha two r and r' is therefore a raotpffooal ono: each of th«a la •• the polar surface of the other. It la easy to mtfy iMh oi u>r^ the particular caaa of tha olroolar halix. 486 SKEW CURVES [XI, §234 Note. It is easy to find the general f ormulse for all skew curves whose radius of curvature is constant. Let R be the given constant radius and let a, /3, y be any three functions of a variable parameter which satisfy the relation a* + /S^ + 72 = l. Then the equations (66) X = U fader, T=Rfpd(r, Z = Rfyd(T, where da = Vda^ + d^ + dy^, represent a curve which has the required prop- erty, and it is easy to show that all curves which have that property may be obtained in this manner. For a, /3, 7 are exactly the direction cosines of the curve defined by (66), and a is the arc of its spherical indicatrix (§ 225). IV. CONTACT BETWEEN SKEW CURVES CONTACT BETWEEN CURVES AND SURFACES 234. Contact between two curves. The order of contact of two skew curves is defined in the same way as for plane curves. Let T and r' be two curves which are tangent at a point A. To each point M of r near A let us assign a point M' of V according to such a law that M and M' approach A simultaneously. We proceed to find the maximum order of the infinitesimal MM' with respect to the principal infinitesimal AM, the arc of T. If this maximum order is n -\- 1, we shall say that the two curves have contact of order n. Let us assume a system of trirectangular * axes in space, such that the yz plane is not parallel to the common tangent at A, and let the equations of the two curves be (F) 1^ = ^(^)' (F') r = ^(^)^ ^ ^ \z=<^(x), ^ ^ \z = ^(x). If a^oj 2/0 ^0 are the coordinates of A, the coordinates of M and ilf ' are, respectively, \xo + h, f(xo + h), <f>(xo + h)-] , [xo + k, F(xo + k), *(xo + A;)] , where A; is a function of h which is defined by" the law of corre- spondence assumed between M and M' and which approaches zero with h. We may select h as the principal infinitesimal instead of the a,TO AM (§ 211) ; and a necessary condition that MM' should be an infinitesimal of order n -f- 1 is that each of the differences k-h, F(x,^k)-f(x,-\-h), ^xo-hk)-<l>(xo + h) • It is easy to show, by passing to the formula for the distance between two points in oblique coordinates, that this assumption is not essential. XI, |2M] CnVTArT ^gj should be an infinitatimal of uidia a 4- 1 or mor«. It foUovi Ikat we must have ♦(«, + *)- ♦(jr. + A)- yA- where a, /3, y remain finite as A approtfihet saia ^t''~^H ^ kv its value h + aA"^* from the first of these eqnalioiis, the latter two become n^ + A + aA-^') -y)[^ + A). /8A-», ♦(a^ + A + aA-*«) - ^a^ + A) - yA-». Expanding F(io + A -f aA-"^*) and ♦(x, 4- A + aA-*«) by Tajlor^s series, all the terms which contain a will have a factor A*^'; l^tuffiit in order that the preceding condition be aatialled, mA oC tte differences ^(^0 + A) -f{x^ + A), ♦(x, + A; - ^x^ + A) should be of order n + 1 or more. It follows that if Jflf' is of order ?i + l* the distance MN between the points Mi and N of IIm two curves which have the same abscissa «^ + A will be at IsasC of order n + 1. Hence the maximum order of tlie iBSaitesiaal ia question will be obtained bj/ putting into c o rrwp on dintt tAs pmrnls of the two curves which have the mtmo ab§eis9n. This maximum order is easily evaluated. Since the two eurvsiB aie tangent we shall have /(xo) = F(Xo), /'(x^) = F'(xO, 4(«W)-*(^). 4Yx^^-#Y«J. Let us suppose for generality that we also hare fXxo) = F"(xo), , /^>(^) - P^(^h but that at least one of the diflereneet does not vanish. Then the distance MM* will be of ovder • «f 1 and the contact will be of order n. This result may also be siMlod as follows : To find the ordsr ^ eontaet ^ Ups mw^tm V mmd P, mm- sider the two teU of jtrofoetums (C, CT) mmd (C„ CQ ^f tks §imm curves on the xy plane and the xa pUmo, ntpwe ii wo lp, mmd /W lAs order of contact of each set; thm lAe mrdmr ef emtmet ^ tkm fhmm curves T and T* vfiU be tho swuUUr ^ Mess Hso. 488 SKEW CURVES [XI, §235 If the two curves r and r' are given in the form (r) x=f{t), y = <t>(t), z = ^(t), (V) X=f{u), Y=^(u), Z = ^(u), they will be tangent at a point u = t = tQii H^o) = <i>(to) , *'(^o) = <f>'(to) , *(^o) = K*o) , ^'(M =^ 'A'(^o) . If we suppose that f'(^o) '^s not zero, the tangent at the point of contact is not parallel to the yz plane, and the points on the two curves which have the same abscissa correspond to the same value of t. In order that the contact should be of order n it is neces- sary and sufficient that each of the infinitesimals ^(t) — <f>(t) and "^(t) — ^(t) should be of order n -\-l with respect to t — t^, i.e. that we should have *'(^o) = «A'(^o), • • •, ^^^XM = «A^"H^o), and that at least one of the differences should not vanish. It is easy to reduce to the preceding the case in which one of the curves V is given by equations of the form (67) x=f{t), y = <f>(t), z = ^{t), and the other curve r' by two implicit equations F{x,y,z) = 0, F^(x,y,z) = 0. Eesuming the reasoning of § 212, we could show that a necessary condition that the contact should be of order ti at a point of F where t=:tQ is that we should have ,gg- (F(<o) = 0, F'(<„) = 0, ..., F«(<„) = 0, ^ -* ^F.(<„) = 0, F!(<o) = 0, •••, Fi">(<o) = 0, where m = nf(t}, *(o> "A(0] . F, (0 = F, if{t), ^(t), i,(t)-] . 235. Osculating curves. Let T be a curve whose equations are given in the form (67), and let V be one of a family of curves in 2n -h 2 parameters a^by c, •', I, which is defined by the equations (69) F{x, y,z,a,b,->-,l) = 0, F, (x, y, z, a, b.Cy^- -, I) = 0. XI, §238] CONTACT lu general it it possible to determino th« 2« 4- 2 [MfimHis bi mok a way that the corresponding ourve V* has ooDUci ol order • vilb the given curve F at a given point The ennre thus dfrtarf jned is called the otculating curve of the familj (69) to the entire T. The equations which determine Uie values of the ptxuietatt «, it i^ • • •, I are precisely the 2m + 2 equations (68). It should be Doled these equations cannot be solved unless eeoh of the fnnntinai F \ h\ contain at least n -f 1 parameters. For example, If the F' are plane curves, one of the equations (69) oontams onlj three parameters; hence a plane curve cannot have eootMi of higher than two with a skew curve at a point takeo at the curve. Let us apply this theory to the simpler dasses of eanreSv — the straight line and the circle. A straight line depends oo foor paia^- eters ; hence the osculating straight line will have oootaei of the first order. It is easy to show that it coincides with the for if we write the equations of the straight line in the form the equations (68) become where (xo, yo, «o) is the supposed point of eootafli on P. Solriaf these equations, we find a = — - < *♦ ■• ^ which are precisely the values which give the taafent A sary condition that the tangent should have eootael of the order is that xH = a«i', y;' = &«;', that is, ri yi H The points where this happens are those disouased in I 21T. The family of all circles in spaoe depends oa six paiMMian; hence the asculaHng eireU will hare oootaei of the eeeood om^. Let the equations of the circle be written in the form F(«,y,«) = v4(«-a) + B(jf-*) + C(e-#) -0. F»(«,y,*) = («-a)« + (y-*)« + («-«)"-^-<>. 490 SKEW CURVES [xi,§236 where the parameters are a, b, c, R, and the two ratios of the three coefficients A, B, C. The equations which determine the osculating circle are A(x - a) + B(y - i) + C(z - c) = 0, (X - ay + (y- by + (z- cy = R\ ^ dx , , ,^dy , , ^dz (x-a)^ + (S,-i)^ + (.-c)^ + f, = 0, where x, y, and z are to be replaced by /(^), </>(^), and j/r(^), respec- tively. The second and the third of these equations show that the plane of the osculating circle is the osculating plane of the curve r. If a, J, and c be thought of as the running coordinates, the last two equations represent, respectively, the normal plane at the point (ic, 2/, «) and the normal plane at a point whose distance from (x, y, «) is infinitesimal. Hence the center of the osculating circle is the point of intersection of the osculating plane and the polar line. It follows that the osculating circle coincides with the circle of curvature, as we might have foreseen by noticing that two curves which have contact of the second order have the same circle of curvature, since the values of y\ z\ y", «" are the same for the two curves. 236. Contact between a curve and a surface. Let 5 be a surface and r a curve tangent to 5 at a point A. To any point M of F near A let us assign a point M' of S according to such a law that M and M' approach A simultaneously. First let us try to find what law of correspondence between M and 3/' will render the order of the infinitesimal MM^ with respect to the arc AM 2u maximum. Let us choose a system of rectangular coordinates in such a way that the tangent to V shall not be parallel to the yz plane, and that the tangent plane to .V shall not be parallel to the z axis. Let (2^0) 2/0? ^c) ^6 the coordinates of ^1 ; Z — F{x, y) the equation of S-, y =/(a;), z = <ti(x) the equations of T ; and w + 1 the order of the infinitesimal MM' for the given law of correspondence. The Xl,5'i3fij CONTACT 491 coordinates of 3/ are [a^ + A, /(as, -h A), ^g^ -♦-*)). Ltl X; Ft Mid ^ = F(A', r) be Uie oo6rdiiuitet of A/'. In otdet tbat i#jr< be of order n -\- 1 with respect to the arc .1 Jl#, or, what the same thing, with respect to A, it U ncirwiir/ that f4 t el Um differences X — x^ Y — y^ and ^ — s should be an iiiftiiiieaiMal ai least of order n + 1> that is, that we should have where a, fiy y remain finite as A approaobaa xero. Heiioa wt shall have F(x 4- aA-^S y + /?*"''*) - « = yA''^*, and the difference F(x, y) — « will be itself at least of order a -f 1. This shows that the order of the infinitesimal MS, where H is tha point where a parallel to the z axis pierces the 8iirfaoa» will bt al least as great as that of MM'. The maximum order of <<^tti*t — which we shall call the order of contaet of the curve and ike mtffmtm — is therefore that of the distance MN with respect to the are AM or with respect to h. Or, again, we may saj that the order of eoih tact of the curve and the surface is the order of eomiaei htiwtem T and the curre T' in which the surface S U cut hy ike ejfUmier wkitk projects r upon the xy plane. (It is erident that the m axis lai^ be any line not parallel to the tangent plane.) For the equatiooa of the curve r' are y=/(x), ;7=F[x,/l[x)] -♦(*), and, by hypothesis, ♦(x„) = <^(ar«). ^'(r,) = ^'(r.). If we also have the curve and the surface have oontaet of order a, Siiiee the eqaa^ tion <P(x) = 4^(x) gives the abeoiassB of the poinU of iateieaetiea of the curve and the surface, these conditions for eootaei of at a point A may be expressed by saying that the onnre surface in n -f- 1 coincident pointa at A. Finally, if the curve T is given by equations of the form » -/^rV. y = 4,(t), z = ^(t), and the surface S is giTen by a sisfle eqnatte of the form F(x, y, «) = 0, the curre T Just defined will have eqM^ tions of the form x ^f{t), y - 4(0. « - »('). ^*>«« •XO *• • '■^ tion defined by the equation 4d2 SKEW CURVES [XI, §237 In order that T and V should have contact of order n, the infini- tesimal 7r(t) — ij/(t) must be of order n -\- 1 with respect to t — to, that is, we must have T(g = 'A(^o) , 7r'(g = if^Xto) , . . • , 7r<»> (to) = V^«> (to) . Using F(^) to denote the function considered in § 234, these equa- tions may be written in the form F(<«) = 0, F'(<„) = 0, ..., F<»'(<„) = 0. These conditions may be expressed by saying that the curve and the surface have n + 1 coincident points of intersection at their point of contact. If S be one of a family of surfaces which depends on n -\-l parameters a, bj c, • • • j I, the parameters may be so chosen that S has contact of order n with a given curve at a given point ; this surface is called the osculating surface. In the case of a plane there are three parameters. The equations which determine these parameters for the osculating plane are Af (t) + B<^ (t) i-Cil; {t)-\-D = 0, Af(t) + B<l>'{t)-\-Cil;'(t) =0, Af"(t) 4- B<t>'Xt) + cr(t) = 0. It is clear that these are the same equations we found before for the osculating plane, and that the contact is in general of the second order. If the order of contact is higher, we must have Af"(t) + B<f>"\t) 4- C,/.'"(0 = 0, i.e. the osculating plane must be stationary. 237. Osculating sphere. The equation of a sphere depends on four parameters; hence the osculating sphere will have contact of the third order. For simplicity let us suppose that the coordinates X, ?/, « of a point of the given curve F are expressed in terms of the arc s of that curve. In order that a sphere whose center is (a, b, c) and whose radius is p should have contact of the third order with r at a given point (x, y, z) on T, we must have F« = 0, F'(5) = 0, F"W = 0, P"(5) = 0, where FW = (X - ay-\-{y - by-\-{z - cy-p^ andwhereaj,y,.M6tipr8MedMfunot40Mart. Esp«MliBt Um la8t three of the eouationi of oonditUm sod applvtiiff PWMf. formulae, we find ''^mmfm F'(«) = («-a)a + (y-.6)/8 + («-c)y-0. ^-<-)-T-'(^7")-'T^(i*f)-4-'a*if) These three equations determine a, 6, and c. Bat the fini of them represents the normal plane to the curve T at the point (», y, c) in the running coordinates («, *, <•)» and the other two maj be deriTcd from this one by differentiating twice with raapect to «. the center of the osculating sphere is the point where the polar li touches its envelope. In order to solve the three aqnattona W9 au^ reduce the last one by means of the others to the form (X - a)a" -f Cy - b)fi" + (* - r)y" = T ^, from which it is easy to derive the formula Hence theradius of the osculating sphere is given by the formttla If R is constant, the oont^r of the osculating sphere ootaeidfie wtth the center of curvature, which agrees with the retnlt obUined la §233. 238. Osculating straight lines. If the eqnatkms of a fiudly td curves depend on n + 2 parameters, the parameCera may be chosen in such a way that the resulting curve C has oontifli of ofder • with a given surface ^^ at a point M. For the equatkm whieh exiirBaBta that C meets S at ^f and the ft + 1 equations which expires thai there are n + 1 coincident points of intefteeiioo ai U n -\-2 equations for the determination of the 494 SKEW CURVES [XI, Exa. For example, the equations of a straight line depend on four parameters. Hence, through each point JW of a given surface 5, there exist one or more straight lines which have contact of the second order with the surface. In order to determine these lines, let us take the origin at the point M, and let us suppose that the z axis is not parallel to the tangent plane at M. Let z — F(xy y) be the equation of the surface with respect to th^se axes. The required line evidently passes through the origin, and its equations are of the form a b c Hence the equation cp = F(ap, hp) should have a triple root p = ; that is, we should have c = ap -{- bq, = a^r-i-2abs-\-bH, where p, q, r, s, t denote the values of the first and second deriva- tives of F{x, y) at the origin. The first of these equations expresses that the required line lies in the tangent plane, which is evident a priori. The second equation is a quadratic equation in the ratio b/a, and its roots are real if s^ — rt is positive. Hence there are in general two and only two straight lines through any point of a given surface which have contact of the second order with that surface. These lines will be real or imaginary according as s^ — r^ is positive or negative. We shall meet these lines again in the following chapter, in the study of the curvature of surfaces. EXERCISES 1. Find, in finite form, the equations of the evolutes of the curve which cuts the straight line generators of a right circular cone at a constant angle. Discuss the problem. [Licence^ Marseilles, July, 1884.] 2. Do there exist skew curves T for which the three points of intersection of a fixed plane P with the tangent, the principal normal, and the binormal are the vertices of an equilateral triangle ? 3. Let r be the edge of regression of a surface which is the envelope of a one-parameter family of spheres, i.e. the envelope of the characteristic circles. Show that the curve which is the locus of the centers of the spheres lies on the polar surface of r. Also state and prove the converse. 4. Let r be a given skew curve, M a point on r, and a fixed point in ipace. Through draw a line parallel to the polar line to r at 3f, and lay off on this parallel a segment O^ equal to the radius of curvature of V at M. Show XI, ExH.] KXKtCIBEi that the curve r' daierlbwl bj Um potet Jiraad tte center of cunrature of r have their tiinnoi length equal, and their radii of eurratura aqual, aft 6. If the oieiilatiiig ipbere lo a givm tkt show that r lies on a iphere of mdiua a, at laaei of r ia constant and equal to a. 6. Show that the neceeeary and foffleiMit «^otKthVm Ihaft Ika center of curvature of a helix drawn on a eyUndtr iboold bt cylinder parallel to the first one is that the right atoikai of Uh should be a circle or a logarithmic iplraL In tha latter CMi i helioea lie on circular cones which hare the same azii and tba [TissoT, NmmUm Ammmlm, Vol XI« lMi.J 7*. If two skew curres have the same prtodpal planes of the two curves at the pohiU where they meei the a constant angle with each other. The two potnta jnet ters of curvature of the two curves form a sjretem of fo«r potote monic ratio is constant. The product of the radii oi tonloa of Um two at corresponding points is a constant. [Paul Raaaar ; llavaHsni ; Scasu.] 8*. Let X, y, z be the rectangular coArdinatce of a poiat <m a rfww c«no P, and 8 the arc of that curve. Then the conre r« deAned bf Ike • Xo-fa"dM, y^sJfiTd*, s»Bj*Y''ds. where xo, Vo, zo &re the running coOrdinatea, li oalled Iho and the curve detined by the eqnatiooe X = xco8tf + xosin^, F = ycos# + ir»ein#, Z » tcos#4- leilit. where X, F, Z are the running ooflcdinatea and # ie a a related curve. Find the orientatioQ of the these curves, and find their radii of conratare and of If the curvature of T is constant, the totiioB of the tnm Ft to the related cunes are corvee of the Bertraiid tjpe (| tIS). general equations of the latter corvee. 9. I.et r and T' be two skew corvee whkh are A lay off infinitesimal arcs ilif and AM' from A aloag the same diroction. Find the limiting podtloa of the Mm Mit 10. In order that a straight line rigidly dron of a skew curve and p ee ring describe a developable sorfaoe, that at least unices the given skew conre to a infinite number of etiaight Unee which hftfO tht 496 SKEW CURVES [XI, Exs. For a curve of the Bertrand type there exist two hyperbolic paraboloids rigidly connected to the fundamental trihedron, each of whose generators describes a developable surface. [CesIro, Rivista di Mathematical Vol. II, 1892, p. 155.] 11*. In order that the principal normals of a given skew curve should be the binormals of another curve, the radii of curvature and the radii of torsion of the first curve must satisfy a relation of the form A where A and B are constants. [Mannheim, Comptes renduSf 1877.] [The case in which a straight line through a point on a skew curve rigidly connected with the fundamental trihedron is also the principal normal (or the binormal) of another skew curve has been discussed by Pellet {Comptes renduSj May, 1887), by Ceskro {Nouvelles Annates, 1888, p. 147), and by Balitrand {Matliesis, 1894, p. 159).] 12. If the osculating plane to a skew curve T is always tangent to a fixed sphere whose center is O, show that the plane through the tangent perpen- dicular to the principal normal passes through O, and show that the ratio of the radius of curvature to the radius of torsion is a linear function of the arc. State and prove the converse theorems. CHAPTER XII SimFACSS I. CURVATURE OF CURVES DRAWN ON ▲ SimPACE 239. Fundamenul formuU. Meusnier*t thtona. In oidar to ttwJT the curvature of a surface ;it a non-singular point 1/, we f>tft H sup- pose the surface referred to a system of rectangulAr ooOidiiialst such that the axis of z is not pai«llel to the tangent plane at M, If the surface is analytic, its equation may be written in the fons where F{x, y) is developable in power series aooording to powets of x — Xq and y — yo in the neighborhood of the point M («^, y,, s,) (§ 194). But the arguments which we shall use do not leqoiie the assumption that the surface should be analytic : we shall owiely suppose that the function F{x, y), together with its first and seeond derivatives, is continuous near the point (jr«, y,) We shaO nss Monge's notation, pt q, r, s, t, for these deriTatiTes. It is seen immediately from the equation of the tangent plane that the direction cosines of the normal to the sorfaoe aie propor- tional to 7>,.^, and — 1. If we adopt as the positive direetioo of the normal that which makes an acute angle with the positive s the actual direction cosines themselTOs A, m r aie girea by formulae -p — y I (2) vrT^MT* vr+^+7« vi+y+ Let C be a curve on the surface S through the point if, the equations of this curve be given in parameter foi functions of the parameter which represent the ooArdinates of n point of this curve satisfy the equation (1), and heno entials satisfy the two relations (3) datnpdx-^-qdy, (4) d^x =:»pd*x -k-qd^y + rds^ + 2*iUJy -f iWjf*. 497 498 SURFACES [XII, §239 The first of these equations means that the tangent to the curve C lies in the tangent plane to the surface. In order to interpret the second geometrically, let us express the differentials which occur in it in terms of known geometrical quantities. If the independent variable be the arc o- of the curve C, we shall have dx dy _ dz _ d'^x _ a' d^y _ /3' d^z _ y' 1^^"^' d^^"' d^~^' d^~R' Io^^r' d^^~R' where the letters a, ft, y, a', p', y', R have the same meanings as in § 229. Substituting these values in (4) and dividing by Vl+^^ + j^, that equation becomes y^ — pa^ — qP^ _ ra^ + '^sa^ + t^'^ R Vl + y + ?' "" Vl+J3^ + ^^ or, by (2), Xa^ + /M)8' + vy' _^ ra^ + 2sap -f t/S^ R ~ vrr^M^' But the numerator Xa' + fip' + vy' is nothing but the cosine of the angle 6 included between the principal normal to C and the positive direction of the normal to the surface ; hence the preceding formule may be written in the form cos e ra^ + 2safi -f tfi^ (5) R Vl+y4-?' This formula is exactly equivalent to the formula (4); hence it contains all the information we can discover concerning the curva- ture of curves drawn on the surface. Since R and Vl -\-p^ -\- q^ are both essentially positive, cos 6 and ra^ + 2sa/3 -f tjS'^ have the same sign, i.e. the sign of the latter quantity shows whether is acute or obtuse. In the first place, let us consider all the curves on the sur- face S through the point M which have the same osculating plane (which shall be other than the tangent plane) at the point M. All these curves have the same tangent, namely the intersection of the osculating plane with the tangent plane to the surface. The direc- tion cosines a, ft, y therefore coincide for all these curves. Again, the principal normal to any of these curves coincides with one of the two directions which can be selected upon the perpendicular to the tangent line in the osculating plane. Let w be the angle which the normal to the surface makes with one of these directions ; then we shall have 6 = 0) or = tt — <d. But the sign of 7-a'^ -h 2saft -f tfi^ shows whether the angle is acute or obtuse ; hence the positive XII, J2»] CURVK.s <>:> A r>LKrAC£ direction of the principal normal i« ihe aame for all Since $ is also the same for all the curvM, the radiua of R is the same for them all ; that is to tay, dY/ rJU rmrwm 0m ik§ «ww face through the poiiU M which have ths mm the same center of curvature. It follows that we need only study the ounralim of the sections of the siirface. First let us study tht TtriMMMi of Ite curvature of the sections of the surface by planes whieh all mm through the same tangent MT, We may suppoee, witiiovl koo of generality, that ra^ + 28a p -f t/J* > 0, for a ohaoge in the dtioette of the z axis is sufficient to change the signs of r, $, and t. For all these plane sections we shall have, therefore, ooi^>0, and tlM angle 6 is acute. If /?| be the radius of ourrateva of the aortinii by the normal plane through 3/r, since the oorreapoodiaf aafle # is zero, we shall have Comparing this formula with equation (5), which giToe the ladiaa of curvature of anv obli(iue seotioOy we find /AN ^ _ooe^ (^> J, '"IT' or R = Ri cos 6, which shows that the eemier ^ m nmimr e •/ mtf obliqm section U the prcjeetum of the emter of rurraimre normal section through the 9ame tawgemi Um$, Tbta it llett»titrr » theorem. ' The preceding theorem reduces the study of the eafralBio of oblique sections to the study of the ourrature of nonuJ ioelkMH. We shall discuss directly the results obtained by Bokr. Flwl Isl us remark that the formula (6) will appear in two differsBl fonM for a normal section according as nr* 4* 2sa/l + I/I* is podtivo or negative. In order to avoid the inoooTOoioDOS of eanyinf %hmB two signs, we shall agree to affix the sign -»- or the sign - to Iho rjulius of curvature /? of a normal soetion aeoordinc as the dlrsette from M to the center of cunrature of the ssetiaii is tho sum as or opposite to the positive direction of the normal to the sorlWts. With this convention, R is given in either ease by tho formula m 1 ra»^2eafi + tf 600 SURFACES [XII, § 239 which shows without ambiguity the direction in which the center of curvature lies. From (7) it is easy to determine the position of the surface with respect to its tangent plane near the point of tangency. For if s^ - rt <0, the quadratic form ra'^ -\- 2sap + tfi^ keeps the same sign — the sign of r and of ^ — as the normal plane turns around the normal; hence all the normal sections have their centers of curvature on the same side of the tangent plane, and therefore all lie on the same side of that plane : the surface is said to be convex at such a point, and the point is called an elliptic point. On the contrary, if s^ — rt > 0, the form ra^ -|- 2sap + tfi^ vanishes for two particular positions of the normal plane, and the corresponding normal sections have, in general, a point of inflection. When the normal plane lies in one of the dihedral angles formed by these two planes, R is positive, and the corresponding section lies above the tan- gent plane ; when the normal plane lies in the other dihedral angle, R is negative, and the section lies below the tangent plane. Hence in this case the surface crosses its tangent plane at the point of tangency. Such a point is called a hyperbolic point. Finally, if s^ — rt = 0, all the normal sections lie on the same side of the tan- gent plane near the point of tangency except that one for which the radius of curvature is infinite. The latter section usually crosses the tangent plane. Such a point is called a parabolic point. It is easy to verify these results by a direct study of the differ- ence n, — z — »' of the values of z for a point on the surface and for the point on the tangent plane at M which projects into the same point (a;, ?/) on the xy plane. For we have z' z=p{x - x^ ^ q{y - y^, whence, for the point of tangency {x^, ?/o)> dx "' 1^ = 0. dy U ;5 = ^ dxdy"^' and dx It follows that if «^ — r^ < 0, w is a maximum or a minimum at M (§ 56), and since n vanishes at M, it has the same sign for all other points in the neighborhood. On the other hand, if 5* — r< > 0, u has neither a maximum nor a minimum at AT, and hence it changes sign in any neighborhood of M. 240. Ettler*! theoremt. The IndioitHx. in onler lo tlody lb« w^ tioii uf the radius of curvaturu of a nomud Molioo, \h m t^km tkt point M as the origin and the tangent pbuie at AT m Um xy f^^m With such a system of axes we shall haTO j» » ^ « o, aad Um formula (7) beoomee (8) ii ~ ** °^**^ + 2f cos 4 sin ^ + < tinV* where <^ is the angle which the trace of the normal plaae with the positive x axis. Equating the derivatiTe of Um inembcr to zero, we find that the pointa at which H maj In a mum or a minimum stand at right angles. The following geomei^ rical picture is a convenient means of visualizing the TariaticMi of M, Let us lay off, on the line of intersection of the normal plane wtlk the xf/ plane, from the origin, a length Om equal numertcallj to Um square root of the absolute value of the corresponding radios of vature. The point m will describe a corrs, which gires an neous picture of the variation of the radius of curratniv. This carra is called the indiccUrix, Let us examine the three possibls oassa. 1) s^ — rt <0. In this case the radius A has a coostaat sign, whleb we shall suppose positive. The coordinates of m are i « VS cos 4 and i; = V^ sin <t> ; hence the equation of the imJitairiM is (9) re-^2iif, + t^^l, which is the equation of an ellipse whose center is the orifia. It is clear that Ji is at a maximum for the section made bf the aocaMl plane through the major axis of this ellipse, and at a miaimam for the normal plane through the minor axis. The sections omde bf two planes which are equally inclined to the two axes eridentljr hare the same curvature. The two sections whose planes pass through the axes of the indicatrix are called the primeipai monmmi m tt im$ t aad the corresponding radii of curvature are called the ptimtipmi mHi ^ curvature. If the axes of the indicatrix are taken for the axas of a and y, we shall have # = 0, and the formula (8) heooHMt 4 = rcosV + <«n*4. n With these axes the principal radii of cvrratare JI, aad if, to <^ = and «^ = 7r/2, respectively; hence l/i», - r, !/£, - I, 1 cos*4 ^ «P*» 502 SURFACES [XII, § 240 2) s^ — rt> 0. The normal sections which correspond to the values of <^ which satisfy the equation r cos^<f> + 2« cos <^ sin <^ + ^ sin^</) = have infinite radii of curvature. Let ZjOZi and L^OZg be the inter- sections of these two planes with the xij plane. When the trace of the normal plane lies in the angle ZjOZg, for example, the radius of curvature is positive. Hence the corresponding portion of the indicatrix is represented by the equation where i and r) are, as in the previous case, the coordinates of the point m. This is an hyperbola whose asymptotes are the lines L[OLi and L^OL^. When the trace of the normal plane lies in the other angle L^OLu R is negative, and the coordinates of m are I = V— R cos <^ , ly = V — i2 sin <^ . Hence the corresponding portion of the indicatrix is the hyperbola re + 2s^'q-\-trf = -l, which is conjugate to the preceding hyperbola. These two hyper- bolas together form a picture of the variation of the radius of curva- ture in this case. If the axes of the hyperbolas be taken as the X and y axes, the formula (8) may be written in the form (10), as in the previous case, where now, however, the principal radii of curva- ture Rx and R^ have opposite signs. 3) s^ — rt = 0. In this case the radius of curvature R has a fixed sign, which we shall suppose positive. The indicatrix is still represented by the equation (9), but, since its center is at the origin and it is of the parabolic type, it must be composed of two parallel straight lines. If the axis of y be taken parallel to these lines, we shall have s = 0, t = Oj and the general formula (8) becomes or 1 _ cos'<^ R" Ri ' This case may also be considered to be a limiting case of either of the preceding, and the formula just found may be thought of as the limiting case of (10), when R^ becomes infinite. XII, 5 241] CURVES ON A 8URPACK Euler'sforaolsmaybettUblUMd wiUuMiftwli«ltolanMk<«|. T^kkm the point M of tbe given mutMM m Um origiii Ami lb* Tinjiiii |iiaM m iJiIiw plane, tbe ezpausion of 1 1^ Taylor*! mHm mi^ bt wHttia lo tW fons l.S ^"'* where the terms not written down are of order grMier 'kft t««. In nidn tu tiiid the radii of curvature of the notion made bj n pUne y ■ a taa a, ve may introduce the transformntion X = s'ooe^ - y'iln^, y = z'lln^ + V'cm^, and then set / = 0. Tble gives the expansion of z in powets of x', ^ _ rcos«^ -t- >ssin»ooa^ ^. f sin«» 1.2 r'^ . . which, by § 214, leads to the formuhi (8) Notes. The section of the surface by iu langeni plane Is given by the eqoalloa = rx« + 2sa5y + fyt + #,(x, r) + • • •. and has a double point at tbe origin. Tbe two tangenU at this potol aie the asymptotic tangents. More generally, if two surfaces 8 and 8% an holh tM^Hl at the origin to the xy plane, the projection of their curve of InlenscUea on iha xy plane is given by the equation = (r - ri)x« + 2(« - $i)xy -^t - <,,yi ^ . . ., where ri, Si, ti have the same meaning for the Aurfaos S% that r, a, f have for S. Tlie nature of the double point depends upon the dfn ol the expceaten (s — «i)> — (r — ri)(t — fi). If this exprenlon ia zrro, the curw uf int Jif— r U ca has, in general, a cusp at the origin. To recapitulate, there exist on any surface four reoiarkmbU pon> tions for the tangeut at any point : two peqwndioular tanceota for which the corresponding radii of curvature have a maxiinttm or a minimum, and two so-<ralled asympioiiej or prtnripal* tangvnta, for which the corresponding radii of curvature are infinite. The latter ara to be found by equating the trinomial ra*4- 2sa/J+ f/J* to aaro (| W^ We proceed to show how to find tlie principal nonaal a ec ii oi n aad the principal radii of curvature for any system of rectangular 241. Principal radii of curvature. There are in geoerml two normal sections wliose radii of curvature are equal to any fives value of R. The only exception is the caee in whieh the five© value of iJ is one of the principal radii of ourvalnre, in which ease • The reader should distinguish sharply the dl wetto as of (the asymptotes of tbe Indlcatrix) and Ihs dlrwjtloas of ^ (thetuccvof the Indicatrlx). To avoid tangent.— Ikkhs. 604 SURFACES [XII, §241 only the conesponding principal section has the assigned radius of curvature. To determine the normal sections whose radius of curvature is a given number R, we may determine the values of a, p, y by the three equations Vl 4- »2 -f </^ ^ = ra'' + 2safi + tjS^ y=pa-\-q/3, a* + ^^ + y' =1 . It is easy to derive from these the following homogeneous combiner tion of degree zero in a and y8 : nU Vl + j>^ + g^ ^ ra^ + 2sal3 + t/S ^ ^ R a' + ^ + {pa^qP) 2 It follows that the ratio p/a is given by the equation a\l + p^ - tD) + 2ap{pq - sD) -^ ^"(1+ q^ - tD) = 0, where R — i) Vl -\-jp^ -\- (f. If this equation has a double root, that root satisfies each of the equations formed by setting the two first derivatives of the left-hand side with respect to a and ^ equal to zero: (12) a{pq-sD)^^{^L-^q^-tD) = 0. Eliminating a and fi and replacing 2) by its value, we obtain an equation for the principal radii of curvature : (13) j (14) {rt-s')R''--\l\^f^q\{\-^p'^t-\-(^-\-q^)r-2pqs'\R + (l+i>' + ?y = 0. On the other hand, eliminating D from the equations (12), we obtain an equation of the second degree which determines the lines of inter- section of the tangent plane with the principal normal sections : a\{l-\- f)s - pqr\ Jrafi\^{\-\-f)t-{\^q^)r-\^f^lpqt-Q.^q^)s-\=.{). From the very nature of the problem the roots of the equations (13) and (14) will surely be real. It is easy to verify this fact directly. In order that the equation for R should have equal roots, it is necessary that the indicatrix should be a circle, in which case all the normal sections will have the same radius of curvature. Hence the second member of (11) must be independent of the ratio j8/a, which necessitates the equations ^^'"'^ 1 + y pq l + ?»* XII. §241] CURVES ON A 8UKFACB 501 The points which satisfy theM equations ar« exiled MwtWw, jU such points the equation (14) roduoat to an idaatilj, alaM •tttj diameter of a circle ia also an axis of ajnuBOlrj. It is often possible to determine the prinetpal from certain geometrical considerationa. For *nffttttim. If a S has a plane of symmetry through a point M od tba tarfaM* H b clear that the line of intersection of that plana with the plane at ^f is a line of symmetry of the indioatrix ; tion by the plane of symmetry is one of the principal example, on a surface of revoluticm the meridian through aay podit is one of the principal normal sections ; it is evident that tlia of the other principal normal sectioo passes through tha the surface and the tangent to the circular parallel at the polat. But we know the center of curvature of one of the obliqi through this tangent line, namely that of the oiroular paralUI U It follows from Meusnier's theorem that the oentar of ciinratiira of the second principal section is the point whara tha nonaal to tkm surface meets the axis of revolution. At any point of a developable surface, «* — H = 0, and tha iadiea- trix is a pair of parallel straight lines. Ona of tha prindpal 8ia> tions coincides with the generator, and tha corresponding radios of curvature is infinite. The plane of the second principal saotion is perpendicular to the generator. All the points of a devalopabla surface are parabolic, and, conversely, thasa are the only siufa aaa which have that property (§ 222). If a non-developable surface is convex at certain points, while oIImt points of the surface are hyperbolic, there is usually a line of pam> bolic points which separates the region where j* - r< is positive froM the region where the same quantity is negative. For eia mp K os thm anchor ring, these parabolic lines are the aztNOM eirenlar p f l M i In general there are on any convex sorfMe only a iafteawiir efaaiMtaa. We proceed to show that the only real MilBee lor wUek trm j p«ii > le m umbilic is the sphere. Let X, m. ' be the direetioa edilB« of lbs aoiaal le Ike surface. Differentiating (2), we find the formote d\_ pq$'-{l + q^r 9X ^ pqi-il-^ ^ » to" (l + pi + g«)« ' *r {X^l^-^^ "■"""• g... g... s-g- 606 SURFACES [XII, § 242 The first equation shows that X is independent of y, the second that fj. is inde- pendent of X ; hence the common value of d\/dx, dfx/dy is independent of both X and y, i.e. it is a constant, say 1/a. This fact leads to the equations . _X-Xo A = » a _y -yo _ Va2 - (X - Xo)2 - a ' a -{y- -J/o)^ P = X X — Xo V y/cC^ -{X- Xo)2 - (y - yo)^ Q = _ A* _ y-yo y Va2 - (X - Xo)2 - (y - yo)2 >, integrating, the value of z is found to be z = zo + v a'-* - (ic - ^oY -{y - yo)'^ , which is the equation of a sphere. It is evident that if d\/dz = dfi/dy = 0, the surface is a plane. But the equations (15) also have an infinite number of imaginary solutions which satisfy the relation I + p^ + q^ = 0, as we can see by differentiating this equation with respect to x and with respect to y. II. ASYMPTOTIC LINES CONJUGATE LINES 242. Definition and properties of asymptotic lines. At every hyper- bolic point of a surface there are two tangents for which the corre- sponding normal sections have infinite radii of curvature, namely the asymptotes of the indicatrix. The curves on the given surface which are tangent at each of their points to one of these asymptotic directions are called asymptotic lines. If a point moves along any curve on a surface, the differentials dx, dy, dz are proportional to the direction cosines of the tangent. For an asymptotic tangent ra^ 4- 2sap -\-tp'^ = 0] hence the differentials dx and dy at any point of an asymptotic line must satisfy the relation (16) rdx^-\-2sdxdy-{-tdy^ = 0. If the equation of the surface be given in the form z = F(x, y), and we substitute for r, s, and t their values as functions of x and y, this equation may be solved for dy/dx, and we shall obtain the two solutions (^^) £=*'(^'j')' 2 =*'(*' 2')- We shall see later that each of these equations has an infinite num- ber of solutions, and that every pair of values (xq, ?/o) determines in general one and only one solution. It follows that there pass through every point of the surface, in general, two and only two Xll,ja42] ASYMPTOTIC LINES rovJi^AiK LOIBI ^ asymptotic lines : all these linee togeUier torm * doohto lines upou the surfaoe. Again, the asymptotio lines may be defined wtthom Iht me el any metrical relation : ths a»ympt4ttie Umm e« « „|,y,Dg ^^ ^j^^^ curves for which the oseuloHng plans alway§ rtimiim wkk Us 9^ gent plane to the tur/aee. For the neoeaaary and siifleieoi eottdilMMi that the osculating plane should ooinoide with the teBgwt dImm to the surface is that the equationa dz —pdx — qdy = (i^ d*z — pd*x — ytPy ^ should be satisfied simultaneously (see f 215). The fif«| of Umm equations is satisfied by any curve which liee oo the torfM. Dif* ferentiating it, we obtain the equation d*z—pd*x — qiPy -dpdx -dqdymO, which shows that the second of the preoeding eqaationfl may be replaced by the following relation between the firit diffeiwitiaU : (18) dpdx-^dqdy^O, an equation which coincides with (16). Motmrm it ia Mij to explain why the two definitions are equivalent Sinoe the radios of curvature of the normal section which is tangent to an asjmptoii of the indicatrix is infinite, the radius of curmtore of Um Mymp* totic line will also be infinite, by Meusnier^i theorem, at leaat unUi the osculating plane is perpendicular to the normal plane, in whkk case Meusnier's theorem becomes illusory. Henee the plane to an asymptotic line must coincide with the taag at least unless the radius of curvature is infinite ; but if thia true, the line would be a straight line and ita oeru l ating would be indeterminate. It follows from this property thai aaj projective transformation carries the asymptotio linea iato MX^P- totic lines. It is evident also that the diffeteiitial e^iakte it of the same form whether the axes are rectangular or ohUqM, for the equation of the osculating plane remains of the smm Hnu It is clear that the asymptotic lines exiat only i» «••• ^ potato of the surface are hyperboUo. But when the tiirlMi It ato^jlta the differential equation (16) always has an infinite noaber of eol»- tious, real or imaginary, whether i" - »f is poailive or neflftivm. As a generalization we shall say that any convex twlMt poitotoM two ey^ tems of imaginary asymptotic lines. Thus the aaynptode liato «f n unparted hyperboloid are the two tyttma of roeliliiiear 508 SURFACES [XII, § 24;{ For an ellipsoid or a sphere these generators are imaginary, but they satisfy the differential equation for the asymptotic lines. Example. Let us try to find the asymptotic lines of the surface z = x"*y*^. In this example we have r = m(m — l)a;'"-2y«, 8 = mnx"^-^y'*-^j < = n(n — l)x™y»-2^ and the differential equation (16) may be written in the form mlm - 1) ( ^ I + 2mn( ^ ) + n(n - 1) = 0. \xdy/ \xdyj This equation may be solved as a quadratic in {ydx)/{xdy). Let hi and ^ be the solutions. Then the two families of asymptotic lines are the curves which project, on the xy plane, into the curves yfii=CiX, yf'i=C2X. 243. Differential equation in parameter form. Let the equations of the surface be given in terms of two parameters u and v : (19) x= f(u, v), y = 4>{u, v), z = il/(uy v) . Using the second definition of asymptotic lines, let us write the equation of the tangent plane in the form (20) A(X-x)-\-B(Y-y)-hC(Z-z) = 0, where A, B, and C satisfy the equations (21) cu cu ou cv cv cv which are the equations for A, B, and C found in § 39. Since the osculating plane of an asymptotic line is the same as this tangent plane, these same coefficients must satisfy the equations Adx +Bdy -\- C dz =0, Ad^x-\-Bd^yhCd^z=0. The first of these equations, as above, is satisfied identically. Differ- entiating it, we see that the second may be replaced by the equation (22) dAdx + dBdy-\-dCdz = 0, which is the required differential equation. If, for example, we set C = — 1 in the equations (21), A and B are equal, respectively, to the partial derivatives p and q oi z with respect to x and y, and the equation (22) coincides with (18). XU,5244] ASYMPTOTIC LINES CONJUGATE UXES £xamp<ef. Ata&«zaaptol«laieoBil<tortJMooDo4dt«#<y/k). VMrnm^ tion 18 equivalent to the (qntom SBii,ysMp,fl. ^(t), m4 y^ tqMlliM Oil beconte These eqaationn are ■atiifled IfweaetCsr^n,^-. »^'(,). j| ;= ^-i^ . *^-^- the equation (22) takee the form One solution of this equation i« o = oonat, whieh givM ih« tors. Dividing by do, the remaining equation ia #^(e)d» _»dM whence the second system of asymptotic lines are the defined by the equation u« = Jir^'(o), which curves -'••© Again, consider the surfaces discnased by Jamst, whose sqoalk» may be written in the form Taking the independent variables t and u = p/x^ the the asymptotic lines may be written in the from which each of the systems of aqrmpCoUc lines may be fooad by a slagls quadrature. A helicoid is a surface defined by equations of the form xspcosM, y = psinw /(^)-fA». The reader may show that the differential eqaation of tbe aqravtotio iMi ll p/"{p)df^ - 2h<Utdp + f^r{p)d»^ « 0, from which u may be found by a single qoadralara. 244. Asymptotic Unee on a mled niifiot. Eliminaiuig A,B,Ukdr between tbe equations (21) and tbe eqnatioQ we find the general differential eqoatioQ of the mfymptolie Mum : dl d± di du du ^ (23) ^ ?* ?* dv dv hr d^x d*y €Pm «0. 510 SURFACES [Xn, §244 This equation does not contain the second differentials <Pu and d'^v, for we have d'^x = ^-fd^u^% dH + ^(ft^2 + 2 ^ dudv\-^^^dv^ and analogous expressions for d!^y and d^z. Subtracting from the third row of the determinant (23) the first row multiplied by d^u and the second row multiplied by d'^v, the differential equation becomes df_ d_^ d_^ du du du dv dv dv ^'/ , , ■^dudv + Y^dv^ cu cv cv^ = 0. Developing this determinant with respect to the elements of the first row and arranging with respect to du and dv, the equation may be written in the form (24) D du^ + 2D'dudv-h D"dv^ = 0, where D, D', and D" denote the three determinants (25) D = dx du dx dv d^ du^ dy^ du djy^ dv a^ du'' d_z du d_z do d^ du^ D" = dx du hi du dz du D 1 dx dv dy dv dz dv d^x d'y d^z dudv dudv dudv dx d^j dz du du du dx dy dz da dv dv ' d^x d'y d^z dv^ dv' du' As an application let us consider a ruled surface, that is, a surface whose equations are of the form x = Xq-^ auj y = yQ + P^h Z = ZQ-\-yU, where Xq, ?/o, Zq, a, )3, y are all functions of a second variable param- eter V. If we set w = 0, the point {xq, yo, Zq) describes a certain curve r which lies on the surface. On the other hand, if we set V = const, and let u vary, the point (x, y, z) will describe a straight- XII, J 9W] ASYMPTOTIC LINES OOXJUQATl Uns 611 line generaU)r of the ruled surface, ftnd the Tmltae at m U uv nam of the line will be proportional to the dietouiee hrtirw tW mIM (Xf y, z) and the point (ar«, y«, ««) at which the g ww iaiw HMeto tiM curve r. It is evident from the formulfi* (25) that #> ■> 0, tlMt V is independent of u, awA tliai />" Im a polynomial of fhf degree in «: Since <fr is a factor of (24), one system of asymplotie lii of the rectilinear generators v ss const DiTidinff by dm^ the ing differential etjuation for the other system of asympCoik Uaea ia of the form (^^^ ~ -f Lu« + Afit + JV « 0, where L, 3/, and N are functions of the single Taiiable «. As eqim* tion of this type possesses certain remarkable propartiei^ wkiell «• shall study later. For example, we shall aee thai lA# ««A«rHeafr ratio of any four solutions is a constanL It followa thai tlie anlMr- monic ratio of the four points in which a generator meets a^y fov asymptotic lines of the other system is the same for all which enables us to discover all the aaymptotic Itnet of the system whenever any three of them are known. We shall also see that whenever one or two integrals of the equatkio (M) aie known, all the rest can be found by two qnadratares or by a aiafle quadrature. Thus, if all the generators meet a fixed stmiflil Uas^ that line' will be an asymptotic line of the aeoood fyftam, aad all the others can be found by two quadraturea. If tlie snrCaee pea- sesses two such rectilinear directrices, we should know two totic lines of the second system, and it would appear that quadrature would be required to find all the othata. Bol w« mm obtain a more complete result. For if a tnrface poaaeaaaa Isfo rectilinear directrices, a projective transformaUoo can be foasd which will carry one of them to infinity and tranafonn the avrfaee into a conoid ; but we saw in f 243 thai the aaymplolk Iteea m a conoid could be found without a single quadratura. 245. Conjugate lines. Any two conjugate diamelara of the todk s trix at a point of a given surface S are called es^afafe To every tangent to the surface there eocreipQwIa a < tangent, which coincides with the fif*i when and only vhen t 512 SURFACES [Xii,§245 tangent is an asymptotic tangent. Let z = F{x^ y) be the equation of the surface 5, and let m and m) be the slopes of the projections of two conjugate tangents on the xy plane. These projections on the xy plane must be harmonic conjugates with respect to the projec- tions of the two asymptotic tangents at the same point of the sur- face. But the slopes of the projections of the asymptotic tangents satisfy the equation r4-2s/Lt + ^/x2 = 0. In order that the projections of the conjugate tangents should be harmonic conjugates with respect to the projections of the asymp- totic tangents, it is necessary and sufficient that we should have (27) r -j- s(??i + m') + tmrn) = 0. If C be a curve on the surface S^ the envelope of the tangent plane to 5 at points along this curve is a developable surface which is tangent to S all along C. At every 'point M of C the generator of this developable is the conjugate tangent to the tangent to C. Along C, X, y, z, p, and q are functions of a single independent variable a. The generator of the developable is defined by the two equations Z-z-p{X-x)-q{Y-y) = 0, — dz -\- p dx -{- q dy — dp(X — x) — dq{Y — y) = 0, the last of which reduces to Y—y_ dp _ rdx + sdy X — X dq sdx-\- tdy Let m be the slope of the projection of the tangent to C and m' the slope of the projection of the generator. Then we shall have dy Y — y , and the preceding equation reduces to the form (27), which proves the theorem stated above. Two one-parameter families of curves on a surface are said to form a conjugate network if the tangents to the two curves of the two families which pass through any point are conjugate tangents at that point. It is evident that there are an infinite number of conjugate networks on any surface, for the first family may be assigried arbitrarily, the second family then being determined by a differential equation of the first order. xn,f24aj ASYMPTOTIC LUrE8 CONJUGATE LINES $%$ Given a Burfaee njpnmniMd by rpiiHrm ol tW j conditions under which Um eortm u » ooml aad t ■ network. If we more Along the eorre t m eoom., i ungent plaru* is repreeented by the two ^ ^ ef tg ne ^(X-x) + B(r-y) + C(£-i)«0. In order that thia atraight Une aboald oolneide with the laoctat to ite tmm u = const., whoee direction coainea are proportional to Ik/Ti, ly/l«, H/H, k is necesaary and aufficieot that we ahoald have DifferenUating the drat of theoe eqoationa with regard to «. we aee thai the second may be replaced by the equation a^ + bI^ + c^ dudv dud* and* 0, and finally the elimination of A^ B, and C between the leads to the necessary and aolBoieiit eondition 0. dudv tucv iMd9 This condition is eqaivalent to Mylag that r, y. t are differential equation of the form (SI) and (V) du 9V du dM du dz dv 9V fv dM d9 d^x r^t ofa (29) where M and N are arbitrary funotiooa of u and e. It ioOoWB thai the kaowl* edge of three distinct integrala of an eqnatloa of ihta fonn le determine the equations of a forfaoe whleh li l ii wi e d to a For example, if we set If = iV = 0, the sum of a function of u am equations are of the form of the a funetlon of t : henee, ob anv aarfMO (30) X =/<«)+/,(•), the curves (u) and (v) form a ooQjogale net' Surfaces of the type (80) may be described in two diflbrent ways by translation such that one of iu poinu IT = ^11)4- ♦li*), « « tH«) + H{9)^ 514 SURFACES [XII, §246 let3fo, Mi^ Mi, Mhe four points of the surface which correspond, respectively, to the four sets of values (wo, Uo), (u, Uo)> (wo, v), (u, v) of the parameters u and u. By (30) these four points are the vertices of a plane parallelogram. If Vq is fixed and u allowed to vary, the point Mi will describe a curve F on the surface ; like- wise, if Mo is kept fixed and v is allowed to vary, the point M2 will describe another curve T' on the surface. It follows that we may generate the surface by giving r a motion of translation which causes the point M^ to describe T', or by giving r' a motion of translation which causes the point Mi to describe T. It is evident from this method of generation that the two families of curves (it) and (v) are conjugate. For example, the tangents to the different positions of T' at the various points of T form a cylinder tangent to the surface along T ; hence the tangents to the two curves at any point are conjugate tangents. III. LINES OF CURVATURE 246. Definition and properties of lines of curvature. A curve on a given surface S is called, a line of curvature if the normals to the surface along that curve form a developable surface. If z =f(x, y) is the equation of the surface referred to a system of rectangular axes, the equations of the normal to the surface are .3^. iX = -pZ + (x+pz), ^ ^ \Y=-qZ -\-{y+qz). The necessary and sufficient condition that this line should describe a developable surface is that the two equations (32) |_ Zdp-{- d(x H- pz) = 0, Zdq-{- d(y -\-qz)=:0 should have a solution in terms of Z (§ 223), that is, that we should have d(x + pz) _ d(y H- q^) dp dq or, more simply, dx -\- pdz _ dy •\- qdz dp dq Again, replacing dz^ dp^ and dq by their values, this equation may be written in the form {\ + p'^)dx+pqdy ^ pqdx + {\-\-q^)dy ^ ^ rdx -\- sdy sdx -\- tdy This equation possesses two solutions in dy/dx which are always real and unequal if the surface is real, except at an umbilic. For, if we replace dx and dy by a and p, respectively, the preceding XII. $246] LINES OP CURVATUU 615 equation coincides with the equation found above [(U), 1 241] tm the determination of the linea of inteneeHoo of Um sriMip*! sonBttl sections with the tangent plane. U follows tbtl tbo timun to U^ lines of curvature through any point ooincido with th» MMm of tW indicatrix. We shall see in the studj of diibroDtaal oqualmt IImI there is one and only one line of curratoro throufh overr WH^ singular point of a surface tangent to eooh ooo of iho nm of tko indicatrix at that point, exoept at ao umbilic. Tbiw Umm aio always real if the surface ia real, and the network wkieli Ihij 1mm is at once orthogonal and conjugate, — a charactericUo pfopottj. Example. Let us determine the linee of oonrataiv ai Um grfti ^r rt irfct g ■ xy/a. In this example u a and the differential etiaation (38) Is (a« + ya)dx« = (a« + x«)dy« or — ~ x- O If we take the positive sign for both radJcalt, the (x + VS^T^Xy + V?nn?) « c. which gives one system of lines of ourratiira. If wi ni (34) XzzxVFT^ + yViiT?. the equation of this system may be writtaa in the lerai X 4. VX« + <i« = C by virtue of the identity (x Vu^TTfl -f y Vx« + aO* + o* = [«r + V(^ -»• ^d^ + ^f It follows that the projections of the lioet of enrvataie of tMi represented by the equation (34), where X Is an arbitrary sliown in the same manner that the proj«etioos of the lines of other system are represented by the equatloo (36) X VFTo* - y VPTi? ■ #. From the equation xy = ac of the gitrwi paraboloid, the n e aU— IS4) aM (35) may be written in the form Vx« + «« + Vy« + «« = C, Vi^ + H-Vy« + «»«C. But the expressions vV + f« and Vf^ + i* npTHMl, tances of the point (x, y, s) from the axes of a aad y. li of curvature on the paraboloid art tkom cufWi /br wkkk Of mm er (ifthediManeetiUfamvpoiMupomthmfnmtktmmnfMnd^Um 51G SURFACES [xn, § 247 247. Evolute of a surface. Let C be a line of curvature on a sur- face S. As a poiut M describes the curve C, the normal MN to the surface remains tangent to a curve r. Let (X, Y, Z) be the coor- dinates of the point A at which MN is tangent to r. The ordinate Z is given by either of the equations (32), which reduce to a single equation since C is a line of curvature. The equations (32) may be written in the form {\-\-p^dx-\-pqdy pq dx-\-{l-\- q^) dy ^ — ^ — ' ~~' • rdx -\- sdy sdx -\- tdy Multiplying each term of the first fraction by dx, each term of the second by dy, and then taking the proportion by composition, we find ^ ^ ^ <^x^ + dy"" + {pdx -^qdyY r dx^ -i- 2sdxdy -^ t dy^ Again, since dx, dy, and dz are proportional to the direction cosines a, ^, y of the tangent, this equation may be written in the form Z -z a' + ^^-h (pec 4- qPY ^ 1 Comparing this formula with (7), which gives the radius of curva- ture R of the normal section tangent to the line of curvature, with the proper sign, we see that it is equivalent to the equation (36) Z-z= ^ where v is the cosine of the acute angle between the z axis and the positive direction of the normal. But « + /?v is exactly the value of Z for the center of curvature of the normal section under con- sideration. It follows that the point of tangency A of the normal MN to its eniielope V coincides with the center of curvature of the principal normal section tangent to C at M. Hence the curve F is the locus of these centers of curvature. If we consider all the lines of curvature of the system to which C belongs, the locus of the cor- responding curves F is a surface 2 to which every normal to the given surface S is tangent. For the normal MN, for example, is tangent at A to the curve F which lies on 2. The other line of curvature C through M cuts C at right angles. The normal to S along C is itself always tangent to a curve F' which is the locus of the centers of curvature of the normal sections xu,§a«] LINES OF CURVATUU •ill tangent to C. The iocus oi this ounre P for all tii» Umm oC ture of the system to whiub C belongs is a surfaoe T to wkkk all the normals to s are Ungent The two surfaiwi 1 and I* aiv aol usually analytically distinct, but form two nappsa of tiM mam av- face, which is then represented by an irredndUa aqnatkm. The normal Ai/i to S is tangent to each of these nappes S aad 1* at tlie two principal centers of curvature A and A* of tiM siiifaai B at the point Af. It is easy to find the tangent planes to the two nappes at the points A and A* (Fig. 51). As the point .U describes the curro C, the normal MN describes the developable surface D whose edge of regression is T; at the same time the point A ' where AfX touches 2' describes a curve y' distinct from T, since the straight line MN cannot remain tangent to twu distinct curves T and V. The developable D iind the surface X are tangent at ^4'; hence the tangent plane to 2' at i4' is tangent to D all along AfN. It follows that it is the plane NMT, which passes through the tangent to C. Similarly, it is evident that the tangent plane to 2 at A is the plane NAfT' through the tan- gent to the other line of curvature C*. The two planes NAfT and NAfT' stand at right leads to the following important conception. Let a normal OM be dropped from any point O in space on the surface S, aad lal A and il' be the principal centers of curvature of S on this BonaaL The tangent planes to S and 2' at ^4 and A', respecltvalj» are pa r paad J B' ular. Since each of these planes passes through Um gives poial C^ it is clear that the two nappes of the 9»ohiie of amy smrfaet S. tk m r r^ from any point O in tpatty appear to cut eaek •CA^r at rifkt amftm . The converse of this proposition will be proved later. W^ St This fact 248. Rodrigues' formolA. If A, /«, r denote the diraettaB of the normal, and A* one of the principal radii of cunralora, th# corresponding principal center of curvature will be given by the formulae (37) X = x^R\, l' = y + /f^, ^-s-f-^r. As the point (x, y, «) describes a Una of enrvature the normal section whose radius of eurvatitre is If, this 518 SURFACES [Xii,§249 curvature, as we have just seen, will describe a curve T tangent to the normal MN ; hence we must have dX_dY_dZ_ or, replacing X, Y, and Z by their values from (37) and omitting the common term dR, dx-\- Rd\ _ dy -\- Rdii _ dz-\- Rdy The value of any of these ratios is zero, for if we take them by composition after multiplying each term of the first ratio by X, of the second by fi, and of the third by v, we obtain another ratio equal to any of the three ; but the denominator of the new ratio is unity, while the numerator \dx ■}- fxdy -\-vdz + R(\ d\ -\- fj, dfi -\- v dv) is identically zero. This gives immediately the formulae of Olinde Rodrigues : (38) dx + Rd\ = 0, di/ + Rdfji = 0, dz-\-Rdv = 0, which are very important in the theory of surfaces. It should be noticed, however, that these formulae apply only to a displacement of the point (x, y, z) along a line of curvature. 249. Lines of curvature in parameter form. If the equations of the surface are given in terms of two parameters u and v in the form (19), the equations of the normal are X-x _ Y-y _ Z -z A ~ B ^ C ' where A, B, and C are determined by the equations (21). The necessary and sufficient condition that this line should describe a developable surface is, by § 223, (39) dx dy dz ABC dA dB dC = 0, where x, y, z, A, B, and C are to be replaced by their expressions in terms of the parameters u and v; hence this is the differential equation of the lines of curvature. Xll,ja«] LTXES OF ilTRVAii RB ^10 As an example let ub nnd lue uii« ul curYAUirc ua iIm Mlsali t saaretao^. s whose equation is equiyalent to the systeia z = pcostf, yspsiaf. i e «#. In this example the equations (or A, B, and C aie -<lcoetf+Biln# = 0, - Aprini -^ Bpemi ^ V*9 Taking C = p, we And ^ = adn «, Jl = - a oosi^. Afier cxpMMtea iication the differential equation (80) becomes dp*-(p« + d^*« = or d0 = ±-^ , Choosing the sign +, for example, and IniegraUnf, w fhid p + V^To* = ««•-•., or p^^l^-S-rHf-M], The projections of these lines of cunrature on the sy piaae are aB are easily constructed. The same method enables us to form the • (if thr tvaad degree for the principal radii of curvature. \'- an;.- ^v r: ]..:• Af B, Cf Xj fi, V we shall have, except for ligti. /* = Va* + B*+C* Vi4M^B*+C« We shall -adopt as the positive direction of the Dormal that vhidi is given by the preceding equations. If i? it a phadpal radiot of curvature, taken with its proper sign, the oo6ttliiiiUt of the eoir^ sponding center of curvature are X^x-^-pA, Y^y-^pB, Z-«+^r, where R^py/A^-^^-k-C; If the point (x, y, t) describes the line of car?alni« Uuif«it to lU principal normal section whose radius of oarraturs b #, w« Iuitv seen that the point (.Y, K, Z) describes a curve T whtefa is t S Bf s a l t'> tlu' i..)rni:il to the surface. Heooo we must have A B 620 SURFACES [XII, § 260 or, denoting the common values of these ratios by dp + A", rdx -\- pdA — AK = 0, (40) \dy-^pdB-BK=0, [dz + pdC - CK = 0. Eliminating p and K from these three equations, we find again the differential equation (39) of the lines of curvature. But if we replace dx, dy, dz, dA, dB, and dC by the expressions dx ^ dx ^ dC ^ dC , w-du + ^-dv, du dv respectively, and then eliminate du, dv, and K, we find an equation for the determination of p : (41) dx dA du " du dx dA dv^f" dv dy ^ dB du ^ du dy ^ dB dv '^ cv dz dC du ^ du dz ^ dC dv^f'dv = 0. If we replace p by r/ ^A'^ + 5^ -}- C^, this equation becomes an equation for the principal radii of curvature. The equations (39) and (41) enable us to answer many questions which we have already considered. For example, the necessary and sufficient condition that a point of a surface should be a para- bolic point is that the coefficient of p^ in (41) should vanish. In order that a point be an umbilic, the equation (39) must be satisfied for all values of du and dv As an example let us find the principal radii of curvature of the rectilinear helicoid. With a slight modification of the notation used above, we shall have in this example x = Mcosu, y= usinu, z = av, A = aeinv, B = — acosv, C = u, and the equation (41) becomes a2p2 _ ^2 _,. ^2^ whence E = ± (a^ + u'^)/a. Hence the principal radii of curvature of the helicoid are numerically equal and opposite in sign. 250. Joachimsthal's theorem. The lines of curvature on certain surfaces may be found by geometrical considerations. For example, it is quite evident that the lines of curvature on a surface of revolu- tion are the meridians and the parallels of the surface, for each of XII, §251] LlNh.^ Ut tLttVATLUK Stl these curves is tangent at erery point to OM of Hm ai« of iW indicatrix at that point This if again ooDfinMd bj IIm that the normals along a meridian form a plmo^ and tiM along a parallel form a oiroular oone, — in eaoh oaM tho form a developable surfaoe. On a developable surface the first system of ham of consists of the generators. The second syttem *»'*Ttfttff of Ike orthogonal trajectories of the generators, that it, of tlM inTnlrtw of the edge of regression (f 231). Theae oan be found bjr a tlifle qitd* rature. If we know one of them, all the reit oan be found vitbom even one quadrature. All of these resolta are eatilj Torifiod direelly. The study of the theory of evolutes of a skew onnre lod Je^ chimsthal to a very important theorem, which is ofUn need in tknft theory. Let S and s' be two surfaoes whoee line of intarteetkn C is a line of curvature on each surface. Tho normal MN to 8 aloiy C describes a developable surface, and the normal MS* to ^ C des(Tibes another developable surface. But each of is normal to C. It follows from f 231 that \f two mrfmtm kmtft m common line of curvature^ they inUrteet at a wmttmmt mm§U mUm$ that line. Conversely, if tiro turfaew ifiUnmft at a esnitanf — ffi^ mmd \f their line of intersection i» a line of e ur va iur $ am am§ ^ f As i , ii if also a line of curvature on the other. For we hare aean tbat if one family of normals to a skew curve C form a doTelopable aurfaee^ the family of normals obtained by turning eaoh of thn finft fa^Hy through the same angle in its normal plane also form a devftlofiahU surface. Any curve whatever on a plane or on a sphere is a lino of cnra^ ture on that surface. It follows as a cocoUary to J< theorem that the necessary and s^ffieient eam diii am thai • or a spherical curve on any susfaee ekamid U a Um ^ mmrnima <t that the plane or the sphere en whiek tMe ettr^e Um ekemid mi tka surface at a constant -angle, 251. Dupln'8 theorem. We have already oontidvid [If 4S, 146] triply orthogonal systems of surfaces. The origin of the tksaiy ol such systems lay in a noted theorem due to Dopin, whieb we sknil proceed to prove : Given any three families qf surfaces wkiek fsrm a tnpiy erikefsmmt system : the intersection <^any tw mgfmm ^ d}§m mt^mikm i§ • line of curvature on each of (Aem. 522 SURFACES [XII, §251 We shall base the proof on the following remark. Let F(x^ y,z)=0 be the equation of a surface tangent to the xi/ plane at the origin. Then we shall have, f or a; = y = « = 0, dF/dx = 0, dFjcy = 0, but dFjdz does not vanish, in general, except when the origin is a singular point It follows that the necessary and sufficient condition that the x and y axes should be the axes of the indicatrix is that s = 0. But the value of this second derivative s = d'^z/dx dy is given by the equation d^F , d^F , d'^F , d^F , dF dx dy dx dz dy dz dz^ dz Since p and q both vanish at the origin, the necessary and sufficient condition that s should vanish there is that we should have d'^F _ ^ ' dxdy Now let the three families of the triply orthogonal system be given by the equations Fii^i Vi ^) = pi, Fi(p^, y, «) = />2, Fs{x, y, «) = p8, where Fi, F2, F^ satisfy the relation ^ ^ dx dx dy dy dz dz and two other similar relations obtained by cyclic permutation of the subscripts 1, 2, 3. Through any point AT in space there passes, in general, one surface of each of the three families. The tangents to the three curves of intersection of these three surfaces form a trirec- tangular trihedron. In order to prove Dupin's theorem, it will be sufficient to show that each of these tangents coincides with one of the axes of the indicatrix on each of the surfaces to which it is tangent. In order to show this, let us take the point M as origin and the edges of the trirectangular trihedron as the axes of coordinates; then the three surfaces pass through the origin tangent, respec- tively, to the three coordinate planes. At the origin we shall have, for example, (S).=«. ©.=»■ ©'<». (ti'». ©).=»• (fii-»- ©.=«■ f^i«»' ©=»• XlLjaoiJ LINKS OF CURVATf'RB The axes of ob and y will be the aut ol \Xm iodtelrii of tW ^(a^» y» ») = at the origin if {i^F^J^ by). • a To is the oaae, let ut differentiate (43) with raspeoi to y, terms which vaniah at the origin ; we find or (44) (SI m." From the two relations analogous to (43) we oouUl dedoM two equations analogous to (44), which may be written down hj cjdie permutation : From (44) and (45) it is evident that we shall have alao (fgi=«. (ai-' m-'' which proves the theorem. A remarkable example of a triply orthogonal syaleai it funiaked by the confocal quadrics diseussed in § 147. It wm dnntHleM llw investigation of this particular system which led Dopia to IIm goh eral theorem. It follows that the lines of cunratnro on an ellipsoid or an hyperboloid (which had been detemuBMl ynwiomiy by Mo^ge) are the lines of intersection of thai tnrCnae with ili The paraboloids represented by thn y-X -2«- A, where A is a variable parameter, form ino li Mf tAyiy system, which determines the lines of eoimtart es IIm Finally, the system discussed in | 346, is triply orthogonal 524 SURFACES [XII, §252 The study of triply orthogonal systems is one of the most interest- ing and one of the most difficult problems of differential geometry. A very large number of memoirs have been published on the subject, the results of which have been collected by Darboux in a recent work.* Any surface ^ belongs to an infinite number of triply orthogonal systems. One of these consists of the family of surfaces parallel to S and the two families of developables formed by the normals along the lines of curvature on S. For, let be any point on the normal MN to the surface S at the point M, and let MT and MT' be the tangents to the two lines of curvature C and C which pass through AI; then the tangent plane to the parallel sur- face through O is parallel to the tangent plane to S at M, and the tangent planes to the two developables described by the normals to S along C and C are the planes MNT and MNT', respectively. These three planes are perpendicular by pairs, which shows that the system is triply orthogonal. An infinite number of triply orthogonal systems can be derived from any ouq known triply orthogonal system by means of succes- sive inversions, since any inversion leaves all angles unchanged. Since any surface whatever is a member of some triply orthogonal system, as we have just seen, it follows that an inversion carries the lines of curvature on any surface over into the lines of curvature on the transformed surface. It is easy to verify this fact directly. 252. Applications to certain classes of surfaces. A large number of problems have been discussed in which it is required to find all the surfaces whose lines of curvature have a preassigned geometrical property. We shall proceed to indicate some of the simpler results. First let us determine all those surfaces for which one system of lines of curvature are circles. By Joachimsthal's theorem, the plane of each of the circles must cut the surface at a constant angle. Hence all the normals to the surface along any circle C of the system must meet the axis of the circle, i.e. the perpendicular to its plane at its center, at the same point 0. The sphere through C about as center is tangent to the surface all along C ; hence the required surface must be the envelope of a one-parameter family of spheres. Conversely, any surface which is the envelope of a one-parameter family of spheres is a solution of the problem, for the characteristic curves, which are circles, evidently form one system of lines of curvature. Surfaces of revolution evidently belong to the preceding class. Another interesting particular case is the so-called tubular surface^ which is the envelope of a sphere of constant radius whose center describes an arbitrary curve r. The characteristic curves are the circles of radius R whose centers lie on V and whose planes are normal to r. The normals to the surface are also normal to r ; * Le<;on8 aur les ayatemea orthogonauz et lea coordonn^ea curvilignea, 1898. Xn,§282] LINES OF LLKVATLKE hence the eeoond lygtem of Uaei of eomtnre aro tko ttntt Is is cat by the developable ■arfMte which nwy be fbratd tnm ilMMfVitoier If both lystema of Udcs of cart»tare on » surfaoe v dirla*, K k diw to^ the preceding argument that the earfaoe may be thooghi of m ike eafiloM «f either of two one-parameter famiUea of ■pheres. LiClH* ^. ^ be «w ^m spheres of the first family, C|, Ct, C| the oorreapoodlag ^^ra ri artilii) im9iL and Mi,Mt,Mtthe three poinu in whieh Ci, Ct, Ct ai«e«t byalteaelcwf^ ture C of the other system. The ipbere 8' which ia t^«i fm C is also tangent to the spheres 5|, ^, fis at Jf, , if,, Jjf,, the required tur/aee is the env€U>pe qf a /aimUif qf ijily mck ^ «4kA three fixed spheres. This surface is the well.jEDOWB D^te cydMs. gave an elegant proof that any Dupin cydide la the mfaee into mhkk * < anchor ring is transformed by a certain invwiioii. Let y is orthogonal to each of the three fixed q>h«rai tfi, Ai, 8%, Aa kt\ pole is a point on the circumference of y earriee tliat etrelt iMo a OCy, and carries the three spheres 0i, 8g, 8t into three ipharai 2tt Se* Zs orthogonal to OCT, that is, the centers of the tiauComMd iphewi lia «• OCT. Let Ci, C^, C^ be the intersecUons of these Wfimm wflk a«y ptaM miniil OO', C a circle tangent to each of the dreles C(. C«, C|, tad S* Ite iplMw on which C is a great circle. It is clear that If renaina tafit to aMb of Ika spheres Zi, 2^, Zg as the whole figure is rerolTed aboot 0<X, aad that the envelope of 2' is an anchor ring whose meridlaB la the eireie C Let us now determine the surface for which all of the liaea of emnuan of one system are plane curves whoee pUuiea are all parallal Let aa laha tka ^ plane parallel to the planes in which theaa Unea of oorvalua lla. aad let zcosar + ysinas Fia^i) be the tangential equation of the eeetioB of the aorfaoe by a panaH to the ay plane, where F(a, z) is a function of a and s which dapaoda « under consideration. The coordinates z and y of a point of given by the preceding equation together with the ai|Ballos dF -zaina + ycoaas -—• 9a The formulsB for z, y, z are (46) z = Fcoear sina, y = Faintf + —ooea. i « s. Any surface may be represented by eqoatloaa of function F{a, z) properly. The only a ioa p tiflM are tfca ^^^^ directing plane is the xy pUne. It is aaay to abow that tho oaaAolaMa A^B^C of the tangent plane may be taken to be il = coaa, Bsdna, C^ ^ hence the cosine of the angle between the nofaal and the t aiia la -F.(a.t) " Vl+ j;(a,a) In order that all the aeotiona by plaaea parmllal to Iha i ture, it is necessary and aoflkient, by 626 SURFACES [XII, §263 these planes cut the surface at a constant angle, i.e. that v be independent of a. This is equivalent to saying that Fz{a^ z) is independent of a, i.e. that F{a^ z) is of the form F{a,z) = <p{z) + rl^{a), where the functions </> and yf/ are arbitrary. Substituting this value in (46), we see that the most general solution of the problem is given by the equations X = yl/{a) cos a — r/{cc) sin a + <f>{z) cos a , (47) -[ 3/ = ^(«) sin a + ^'(a) cos a: + <f>{z) sin a , fx = \f/{a) cos a — yf/{ -j 3/ = ^(«) sin a + ^'( These surfaces may be generated as follows. The first two of equations (47), for z constant and a variable, represent a family of parallel curves which are the projections on the xy plane of the sections of the surface by planes parallel to the xy plane. But these curves are all parallel to the curve obtained by set- ting <p{z) = 0. Hence the surfaces may be generated as follows : Taking in the xy plane any curve whatever and its parallel curves^ lift each of the curves verti- cally a distance given by some arbitrary law ; the curves in their new positions form a surface which is the most general solution of the problem. It is easy to see that the preceding construction may be replaced by the following : The required surfaces are those described by any plane curve whose plane rolls without slipping on a cylinder of any base. By analogy with plane curves, these surfaces may be called rolled surfaces or roulettes. This fact may be verified by examining the plane curves a = const. The two families of lines of curvature are the plane curves z = const, and a = const. IV. FAMILIES OF STRAIGHT LINES The equations of a straight line in space contain four variable parameters. Hence we may consider one-, two-, or three-parameter families of straight lines, according to the number of given relations between the four parameters. A one-parameter family of straight lines form a ruled surface. A two-parameter family of straight lines is called a line congruence, and, finally, a three-parameter family of straight lines is called a line complex. 253. Ruled surfaces. Let the equations of a one-parameter family of straight lines (G)he given in the form (48) x = az+pj y=:bz-{-qy where a, ft, jo, q are functions of a single variable parameter u. Let us consider the variation in the position of the tangent plane to the surface S formed by these lines as the point of tangency moves along any one of the generators G. The equations (48), together with the equation z = z, give the cobrdinates x, y, zot & point M on S in terms Xll.fgtt] FAMILIES OF STRAIGHT LUiM^ of the two parametert t ud u ; h«iio«, bj f M^ tb* •^mMm of tk§ tftDgent plane at Af U X-x r-y Z-« -0, where a\ 6', />', q' denote the deriTatiTM of m^h,p,f vtUi to u. Replacing x and y by <» + /> and M + f , simplifying, this equation beoomet (49) (b'z + q')(X - aZ ^p) - (a'« +/»')(K - *Z - f) - 0. In the first place, We see that this plana always passw tliroofk tlM generator G^ which was erident a priarif and moreorer, tlial Um plaA« turns around G as the point of tangenoy At mores aloof G, ai WmI unless the ratio (a'x + p')/{h'x + g') is indepsodoDt of t, i^. uiteia a';' — d'^' = 0, — we shall discard this special ease in what folkmt. Since the preceding ratio is linear in s, eTOty plane throofh a pm* erator is tangent to the surface at one and only ooa point As tbt point of tangenoy recedes indefinitely along the gao«ralor in oHbar direction the tangent plane P approaches a limiting poatUoft /**, which we shall call the tangent plane at the poini ai it\^miif OB tlMI generator. The equation of this limiting piano P* is (60) b\X - oZ - /») - a'(r - *Z - f ) - 0. Let o> be the angle between this plane P* and the tanftnt plaao P al a point M (x, y, z) of the generator. The diroetioo ootiiMS («', /T, /^ and (a, ft y) of the normals to P' and /* are proportional lo b\ -a\ o'4-aA' and *'« + ?', -(a'«4-p'), 6(«'»+/>')-«i,<'«^?'). respectiyely ; hence cos « = aa- + /J/r + ry' - ;^v5i== • where After an easy reduction, we find, by Lagrange's identity (1 15tV (51) tan« = ^j;^:p^-^-«^ ^^ 528 SURFACES [XII, §253 It follows that the limiting plane P' is perpendicular to the tangent plane Pi at a point Oi of the generator whose ordinate Zi is given by the formula xroN ^ _ ay + b'q' + (ab' - ba^jaq' - bp') ^^ ' A a'^ + b''' + {ab'-ba'y The point Oi is called the central point of the generator, and the tan- gent plane Pj at Oi is called the central plane. The angle $ between the tangent plane P at any point M of the generator and this central plane Pi is 7r/2 — <o, and the formula (51) may be replaced by the formula ^,. ^ - _l (l:ziil - r^" + b'^ + (g^- - ba^Mz - gi2, V^ C - ^2 (^f^r _ jr^f) Vl + a2 -f ^2 Let p be the distance between the central point Oi and the point My taken with the sign + or the sign — according as the angle which OiM makes with the positive z axis is acute or obtuse. Then we shall have p = (z — z^) Vl -f a* + b^, and the preceding formula may be written in the form (53) tan^ = A:p, where k, which is called the parameter of distribution^ is defined by the equation _ a^'^^h^^ + jab^ -bay ^ ^ ~ (a'^' - b^p%l + a=^ 4- b^) The formula (53) expresses in very simple form the manner in which the tangent plane turns about the generator. It contains no quantity which does not have a geometrical meaning : we shall see presently that k may be defined geometrically. However, there remains a cer- tain ambiguity in the formula (53), for it is not immediately evident in which sense the angle 6 should be counted. In other words, it is not clear, a priori^ in which direction the tangent plane turns around the generator as the point moves along the generator. The sense of this rotation may be determined by the sign of k. In order to see the matter clearly, imagine an observer lying on a generator G. As the point of tangency M moves from his feet toward his head he will see the tangent plane P turn either from hi